POLYNOMIAL SOLUTIONS OF PELL S EQUATION AND FUNDAMENTAL UNITS IN REAL QUADRATIC FIELDS

Size: px
Start display at page:

Download "POLYNOMIAL SOLUTIONS OF PELL S EQUATION AND FUNDAMENTAL UNITS IN REAL QUADRATIC FIELDS"

Transcription

1 J. London Math. Soc. 67 (2003) C 2003 London Mathematical Society DOI: /S X POLYNOMIAL SOLUTIONS OF PELL S EQUATION AND FUNDAMENTAL UNITS IN REAL QUADRATIC FIELDS J. MCLAUGHLIN Abstract Finding polynomial solutions of Pell s equation is of interest as such solutions sometimes allow the fundamental units to be determined in an infinite class of real quadratic fields. In the paper, for each triple of positive integers (c, h, f) satisfying c 2 fh 2 = 1, where (c, h) are the smallest pair of integers satisfying this equation, several sets of polynomials (c(t),h(t),f(t)) that satisfy c(t) 2 f(t)h(t) 2 = 1 and (c(0),h(0), f(0)) = (c, h, f) are derived. Moreover, it is shown that the pair (c(t),h(t)) constitute the fundamental polynomial solution to the Pell equation above. The continued fraction expansion of f(t) is given in certain general cases (for example when the continued fraction expansion of f has odd period length, or has even period length, or has period length 2 mod 4 and the middle quotient has a particular form, etc.). Some applications to the determination of the fundamental unit in real quadratic fields is also discussed. 1. Introduction: polynomial solutions of Pell s equation Finding polynomial solutions of Pell s equation is of interest as such solutions sometimes allow the fundamental units to be determined in an infinite class of real quadratic fields. Finding such polynomial solutions is not only an interesting problem in its own right; if the fundamental unit in a real quadratic field is known, this means that the class number of the field can be found via some version of Dirichlet s class number formula (see, for example, [12]). In [10], Perron gives some examples of polynomials f for which the continued fraction expansion of f can be stated explicitly, but the period in these cases is at most 6. These examples were added to in a paper by Yamamoto [17]. Later, in [1], Bernstein obtained a large number of such polynomials f for which the continued fraction expansion of f could be made arbitrarily long, and in [2] he gave explicit expressions for the fundamental unit in the quadratic field ( f). Further examples were given in papers by Levesque and Rhin [5], Madden [6], van der Poorten [15] and van der Poorten and Williams [16]. In this paper, for each triple of positive integers (c, h, f) satisfying c 2 fh 2 =1, where (c, h) are the smallest pair of integers satisfying this equation, several sets of polynomials (c(t),h(t),f(t)) that satisfy c(t) 2 f(t)h(t) 2 = 1 and (c(0),h(0),f(0)) = (c, h, f) are derived. Moreover, it is shown that the pair (c(t), h(t)) constitute the fundamental polynomial solution to the Pell equation above. The continued fraction expansion of f(t) is given in certain general cases (for example when the continued fraction expansion of f has odd period length, or Received 24 March 2001; revised 1 March Mathematics Subject Classification 11A55 (primary).

2 polynomial solutions of pell s equation 17 has even period length, or has period length 2 mod 4 and the middle quotient has a particular form, etc.). Some applications to the determination of the fundamental unit in real quadratic fields are also discussed. Definition 1. A polynomial f(t) Z[t] is called a Fermat Pell polynomial in one variable if there exist polynomials c(t),h(t) Z[t] such that c(t) 2 f(t)h(t) 2 = 1 for all t. (1.1) A triple of polynomials (c(t), h(t), f(t)) satisfying equation (1.1) constitutes a polynomial solution to Pell s equation. A Fermat Pell polynomial is said to have a polynomial continued fraction expansion if there exist an integer T and a positive integer n such that f(t) =[a0 (t); a 1 (t),...,a n (t)], (1.2) for all integral t T, where the a i (t) [t]. For a fixed Fermat Pell polynomial f(t), a pair of polynomials (c(t), h(t)) is said to be the fundamental polynomial solution of 1.1 if (c(t), h(t)) constitutes the smallest solution in positive integers to equation (1.1) for all non-negative integral values of t T. This paper shows how to construct several infinite families of Fermat Pell polynomials in which the continued fraction expansion of the polynomials can have arbitrary long period length. Moreover, the polynomial continued fraction expansion of these polynomials can be written down explicitly. 2. Notation for and useful facts about convergents As usual, [a 0 ; a 1,...,a n, 2a 0 ] will denote the simple infinite periodic continued fraction 1 a a an a n + 1 2a 0 + a A i will denote the ith convergent: B i 1 a a 1 + a a i Use will also be made of the following: A i = a i A i 1 + A i 2 B i = a i B i 1 + B i 2 (2.1) A i B i 1 A i 1 B i =( 1) i 1

3 18 j. mclaughlin where A 1 = B 2 =1, A 2 = B 1 =0, each of these being valid for i =0, 1, 2, Some necessary lemmas Before the main results of the paper are given, several lemmas from the theory of the convergents of the continued fraction expansion of f, where f is a non-square positive integer, are needed. Some are given without proof, since the results are well known or are straightforward. In what follows, c and h denote the smallest pair of positive integers satisfying c 2 fh 2 = 1. Recall that, if the continued fraction expansion of f has period 2m, then (c, h) =(A 2m 1,B 2m 1 ), and that if the continued fraction expansion of f has period 2m + 1, then (c, h) =(A 4m+1,B 4m+1 ). by Lemma 1. Then Denote the ith convergent to the continued fraction b b 1 + b b b 1 + b = P i. b i Q i b m b m 1 + b m = P m for all m 1. (3.1) b 0 P m 1 The proof of this lemma is well known. Label the usual three sequences of positive integers used to determine the continued fraction expansion [a 0 ; a 1,...]of f as follows: r 0 =0,s 0 =1,a 0 = f, r k+1 = a k s k r k, s k+1 =(f r 2 k+1 )/s k and a k+1 = ( f + r k+1 )/s k+1,fork 0. Lemma 2. A k A k 1 fb k B k 1 =( 1) k r k+1 k 1 (3.2) A 2 k fbk 2 =( 1) k+1 s k+1 k 1. (3.3) Proof. The lemma is proved through a simple double induction. Lemma 3. r k a 0 for all k 0. Proof. r k+1 = a k s k r k = ( f + r k )/s k s k r k < (( f + r k )/s k )s k r k = f<a Lemma 4. If the continued fraction expansion of f has period 2m and a m = a 0 or a 0 1, then s m =2.

4 polynomial solutions of pell s equation 19 Proof. By Lemma 3 and the definition of a 0, ( f + rm ) a m = < ( f + r m ) < (2a 0 +1). s m s m s m If a m = a 0, then a 0 < (2a 0 +1)/s m and the fact that r m 0 clearly imply that s m =2 (and r m = a 0 ). The case for a m = a 0 1 is similar (with r m = a 0 1). Lemma 5. If the continued fraction expansion of f has period of even length, say equal to 2m, then c = A m B m 1 + A m 1 B m 2, h = B m 1 (B m + B m 2 ). Proof. By the symmetry of the continued fraction expansion of f, [a 1,a 2,...,a 2m 2,a 2m 1 ]=[a 1,a 2,...,a m 1,a m,a m 1,...,a 2,a 1 ]. We know that c = A 2m 1, h = B 2m 1, and, by the correspondence between matrices and convergents, ( )( ) ( ) ( ) a0 1 a1 1 a2m 1 1 A2m 1 A... = 2m B 2m 1 B 2m 2 ( )( ) ( ) ( ) a0 1 a1 1 am 1 Am A... = m B m B m 1 ( ) ( ) ( ) am 1 1 a1 1 Bm 1 A... = m 1 a 0 B m 1, B m 2 A m 2 a 0 B m 2 so that ( ) ( )( ) A2m 1 A 2m 2 Am A = m 1 Bm 1 A m 1 a 0 B m 1, B 2m 1 B 2m 2 B m B m 1 B m 2 A m 2 a 0 B m 2 and the result follows. Remark 1. One can show in essentially the same way that, if the period of the continued fraction expansion of f is odd, say 2m + 1, then c = A 4m+1 =(A 2 m + A 2 m 1)(Bm 2 + Bm 1)+(A 2 m B m + A m 1 B m 1 ) 2 =2(A m B m + A m 1 B m 1 ) 2 +1, h = B 4m+1 =2(A m B m + A m 1 B m 1 )(Bm 2 + Bm 1). 2 Lemma 6. If the continued fraction expansion of f has period of even length that is congruent to 2 mod 4, say equal to 2m, then Proof. By Lemma 5, ha m 1 (c 1)B m 1 =0. ha m 1 (c 1)B m 1 = B m 1 (B m + B m 2 )A m 1 (A m B m 1 + A m 1 B m 2 1)B m 1 = B m 1 (a m B m 1 + B m 2 + B m 2 )A m 1 ((a m A m 1 + A m 2 )B m 1 + A m 1 B m 2 1)B m 1

5 20 j. mclaughlin =2A m 1 B m 1 B m 2 (A m 2 Bm A m 1 B m 1 B m 2 B m 1 ) = B m 1 (A m 1 B m 2 A m 2 B m 1 +1) = B m 1 (( 1) m 2 +1)=0. Lemma 7. If f =[a 0 ; a 1,...,a n, 2a 0 ], then A n = a 0 B n + B n 1. Proof. Case 1 (the period is even, 2m say): By Lemma 5, A n = A m B m 1 + A m 1 B m 2 B n = B m 1 (B m + B m 2 ) B n 1 = B m (A m 1 a 0 B m 1 )+B m 1 (A m 2 a 0 B m 2 ), and the result follows easily. Case 2 (the period is odd): Suppose that the period is odd, 2m + 1 say, and f =[a0 ; a 1,...,a m,a m,...,a 1, 2a 0 ]. Then ( An A n 1 B n B n 1 ) ( Am A = m 1 B m B m 1 )( Bm A m a 0 B m B m 1 A m 1 a 0 B m 1 ) A n = A m B m + A m 1 B m 1 B n = Bm 2 + Bm 1 2 B n 1 = B m (A m a 0 B m )+B m 1 (A m 1 a 0 B m 1 ), and again the result follows by simple arithmetic. Lemma 8. If f =[a 0 ; a 1,...,a n, 2a 0 ], then B n (f a 2 0 )=A n 1 + a 0 B n 1. Proof. Recall that if the period is even (n odd), then A 2 n fbn 2 = 1, that if the period is odd (n even), then A 2 n fbn 2 = 1, and that A n B n 1 A n 1 B n =( 1) n 1. Then B n (f a 2 0 )=A n 1 + a 0 B n 1 B n f a 0 (A n B n 1 )=A n 1 + a 0 B n 1 (by Lemma 7) B n f = a 0 A n + A n 1 Bnf 2 = a 0 A n B n + A n 1 B n A 2 n +( 1) n = a 0 A n B n + A n B n 1 +( 1) n A n = a 0 B n + B n 1, and the result is true by Lemma 7. Lemma 9. If the continued fraction expansion of f has period of odd length, 2m +1 say, then c =2A 2 2m +1 and h =2A 2mB 2m. The proof of this lemma is elementary. Lemma 10. Let f =[a0 ; a 1,...,a m 1,a m,a m 1,...,a 1, 2a 0 ], where a m = a 0 or a 0 1. Then

6 polynomial solutions of pell s equation 21 (i) A m 1 = B m + B m 2 ; (ii) c 1=A m 1 (B m + B m 2 ), if m is odd; (iii) c +1=A m 1 (B m + B m 2 ), if m is even. Proof. (i) By the symmetry of the sequence {r i } (r m+i = r m i+1 ), it follows that r m+1 = r m. By Lemma 2 and Lemma 4, A m A m 1 fb m B m 1 =( 1) m r m+1 =( 1) m a m A 2 m 1 fb2 m 1 =( 1)m s m =2( 1) m = f = A2 m 1 +( 1)m 1 2 Bm 1 2 = A ma m 1 +( 1) m 1 a m B m B m 1 = A 2 m 1 B m +( 1) m 1 2B m = A m A m 1 B m 1 +( 1) m 1 a m B m 1 = A 2 m 1 B m +2B m = A m 1 (A m 1 B m +( 1) m 1 )+a m B m 1 = 2B m = A m 1 + a m B m 1. The result follows from the recurrence relation for the B i. (ii) By Lemma 5, c 1=A m B m 1 + A m 1 B m 2 1 = A m 1 B m +( 1) m 1 + A m 1 B m 2 1 = A m 1 B m + A m 1 B m 2. (iii) Again by Lemma 5, c +1=A m B m 1 + A m 1 B m 2 +1 = A m 1 B m +( 1) m 1 + A m 1 B m 2 +1 = A m 1 B m + A m 1 B m 2. Lemma 11. Let f =[a 0 ; a 1,...,a n, 2a 0 ].IfX and Y are positive integers satisfying X 2 fy 2 = ±1, then (X,Y )=(A k(n+1) 1,B k(n+1) 1 ) for some positive integer k. The proof of this lemma is well known. (See, for example, [3, p. 387]). The way in which Lemma 11 will be used is as follows. Suppose that it is known that X 2 fy 2 = ±1, and that the continued fraction expansion of X/Y is [a 0 ; a 1,...,a n ]. By the lemma, X/Y = A k(n+1) 1 /B k(n+1) 1 for some positive integer k, and thus the finite continued fraction expansion [a 0 ; a 1,...,a n ] either contains at least one complete period of the continued fraction expansion of f (the case k>1), or else is just one term short of a full period (the case k = 1), in which case the missing term is of course 2a 0 and f =[a 0 ; a 1,...,a n, 2a 0 ]. In particular, if a i 2a 0 for some i, 1 i n, then the sequence a 0,a 1,...,a n cannot contain a full period in the continued fraction expansion of f (since a full period of course ends in the term 2a 0 ). This implies that k = 1 and that the latter case holds. In the following theorems, f =[a 0 ; a 1,...,a n, 2a 0 ], unless otherwise stated. As above, c and h are the smallest pair of positive integers satisfying c 2 fh 2 = 1. The variable t will be assumed to be a real variable. Initially, in the theorems that follow, the results will be shown to be true for non-negative integral t and follow for real t by continuation.

7 22 j. mclaughlin Theorem 1. Let f(t) =h 2 t 2 +2ct + f. (i) If n is even, then f(t) =[ht + a0 ; a 1,...,a n, 2a 0,a 1,...,a n, 2(ht + a 0 )] for all t 0. (ii) If n is odd, then f(t) =[ht + a0 ; a 1,...,a n, 2(ht + a 0 )] for all t 0. (iii) In either case, X = h 2 t + c, Y = h constitute the fundamental solution of X 2 f(t)y 2 =1. Proof. It can be easily checked that (h 2 t + c) 2 f(t)h 2 =1. (i) For n even, [ [ht + a 0 ; a 1,...,a n, 2a 0,a 1,...,a n ]= ht + a 0 ; a 1,...,a n,a 0 + A ] n B ( ) n a 0 + A n B n A n + A n 1 = ht + ( ) a 0 + A n B n B n + B n 1 = ht + (a 0B n + A n ) A n + A n 1 B n (a 0 B n + A n )B n + B n 1 B n = ht + 2A2 n 1 (by Lemma 7 and 2.1) 2A n B n = ht + c h = h2 t + c (by Lemma 9). h Since 2(ht + a 0 ) is not in the sequence {a 1,...,a n, 2a 0,a 1,...,a n }, it follows from Lemma 11 that f(t) has the form claimed and that (iii) holds in the case when n is even. (ii) The case for n odd follows similarly, since by a remark preceding Lemma 1. [ht + a 0 ; a 1,...,a n ]=ht + A n /B n = ht + c/h, Theorem 2. Let f(t) =(c 1) 2 h 2 t 2 +2(c 1) 2 t + f. (i) If n is even, then f(t) =[(c 1)ht + a0 ; a 1,...,a n, 2(c 1)ht +2a 0 ]. (ii) If n is odd and f =[a0 ; a 1,...,a m 1,a m,a m 1,...,a 1, 2a 0 ], where a m = a 0 or a 0 1 and m is odd, then f(t) =[(c 1)ht +a 0 ; a 1,...,a m 1, (c 1)ht + a m,a m 1,...,a 1, 2(c 1)ht +2a 0 ]. (iii) In either case, X =(c 1)h 4 t 2 +2(c 1)h 2 t + c, Y = h 3 t + h constitute the fundamental solution of X 2 f(t)y 2 =1. Proof. In (iii), straightforward calculation shows that the expressions given for

8 polynomial solutions of pell s equation 23 X and Y do constitute a solution to X 2 f(t)y 2 = 1. What needs to be shown for (iii) is that these choices of X and Y give the fundamental solution. (i) Recall that, for n even, A 2 n fbn 2 = 1. Notice that [(c 1)ht + a 0 ; a 1,...,a n ]=(c 1)ht + A n = (c 1)htB n + A n, B n B n ((c 1)htB n + A n ) 2 f(t)bn 2 = 1, by the remark above, the formula for f(t) and Lemma 9. Therefore, by Lemma 11, f(t) has the form claimed and, by Lemma 9, the smallest solution to X 2 f(t)y 2 =1 is given by X = 2((c 1)htB n + A n ) 2 +1=(c 1)h 4 t 2 +2(c 1)h 2 t + c Y = 2((c 1)htB n + A n )B n = h 3 t + h. (ii) For n odd, [(c 1)ht + a 0 ; a 1,...,a m 1, (c 1)ht + a m,a m 1,...,a 1 ] [ = (c 1)ht + a 0 ; a 1,...,a m 1, (c 1)ht + B ] m (by Lemma 1) B m 1 ( (c 1)ht + B ) m A m 1 + A m 2 B m 1 =(c 1)ht + ( (c 1)ht + B ) m B m 1 + B m 2 B m 1 =(c 1)ht + (c 1)htB m 1A m 1 + B m B m 1 + A m 2 B m 1 (c 1)htB 2 m 1 + B mb m 1 + B m 2 B m 1 =(c 1)ht + (c 1)h2 t + c h 3 t + h (by Lemma 5, Lemma 10 and 2.1) = (c 1)h4 t 2 +2(c 1)h 2 t + c h 3. t + h The results follow by Lemma 11. Theorem 3. Let f(t) =(c +1) 2 h 2 t 2 +2(c +1) 2 t + f. Ifn is odd and f =[a0 ; a 1,...,a m 1,a m,a m 1,...,a 1, 2a 0 ], where a m = a 0 or a 0 1 and m is even, then f(t) =[(c +1)ht +a 0 ; a 1,...,a m 1, (c +1)ht + a m,a m 1,...,a 1, 2(c +1)ht +2a 0 ] and X =(c +1)h 4 t 2 +2(c +1)h 2 t + c, Y = h 3 t + h constitute the fundamental solution of X 2 f(t)y 2 =1. Proof. The proof here is virtually identical to the proof of Theorem 2(ii), the only difference being that Lemma 10(iii) is used instead of Lemma 10(ii). Theorem 4. Let f(t) =(c +1) 2 h 2 t 2 +2(c 2 1)t + f. Ifn is even, then f(t) =[(c +1)ht + a0 ; a 1,...,a n, 2(c +1)ht +2a 0 ]

9 24 j. mclaughlin and (c +1)2 X = c 1 h4 t 2 +2(c +1)h 2 t + c, Y = constitute the fundamental solution to X 2 f(t)y 2 =1. (c +1) c 1 h3 t + h Proof. By Lemma 9, c =2A 2 n + 1 and h =2A n B n, and so Bn 2 = h 2 /(2c 2). Also, A 2 n fbn 2 = 1. [(c +1)ht + a 0 ; a 1,...,a n ]=(c +1)ht + A n = (c +1)htB n + A n, B n B n ((c +1)htB n + A n ) 2 f(t)bn 2 =2tB n (c + 1)(A n h (c 1)B n ) 1 = 1, by Lemma 9. Therefore the continued fraction expansion of f(t) has the form claimed, and, again by Lemma 9, the fundamental solution of X 2 f(t)y 2 =1 is given by X = 2((c +1)htB n + A n ) 2 +1 =2(c +1) 2 h 2 t 2 B 2 n +4(c +1)htB n A n +2A 2 n +1 = (c +1)2 c 1 h4 t 2 +2(c +1)h 2 t + c, Y = 2((c +1)htB n + A n )B n =2(c +1)htB 2 n +2A n B n = (c +1) (c 1) h3 t + h. Theorem 5. Let f(t) =(c 1) 2 h 6 t 4 +4(c 1) 2 h 4 t 3 +6(c 1) 2 h 2 t 2 +2(c 1)(2c 1)t + f. (i) If n is odd and f =[a0 ; a 1,...,a m 1,a m,a m 1,...,a 1, 2a 0 ], where a m = a 0 or a 0 1 and m is odd, then f(t) =[(c 1)(h 3 t 2 +2ht) +a 0 ; a 1,...,a m 1, (c 1)ht + a m,a m 1,...,a 1, 2(c 1)(h 3 t 2 +2ht)+2a 0 ]. (ii) If n is even, then f(t) =[(c 1)(h 3 t 2 +2ht) +a 0 ; a 1,...,a n, 2(c 1)ht +2a 0,a 1,...,a n, 2(c 1)(h 3 t 2 +2ht)+2a 0 ]. (iii) In either case, X =(c 1)h 6 t 3 +3(c 1)h 4 t 2 +3(c 1)h 2 t + c, Y = h + h 3 t constitute the fundamental solution of X 2 f(t)y 2 =1. Proof. (i) As in the proof of Theorem 2, [(c 1)(h 3 t 2 +2ht)+a 0 ; a 1,...,a m 1, (c 1)ht + a m,a m 1,...,a 1 ] =(c 1)(h 3 t 2 +2ht)+ (c 1)h2 t + c h 3 t + h = (c 1)h6 t 3 +3(c 1)h 4 t 2 +3(c 1)h 2 t + c h 3. t + h

10 polynomial solutions of pell s equation 25 The results follow from Lemma 11. (ii) Similarly, [(c 1)(h 3 t 2 +2ht)+a 0 ; a 1,...,a n, 2(c 1)ht +2a 0,a 1,...,a n ] [ = (c 1)(h 3 t 2 +2ht)+a 0 ; a 1,...,a n, 2(c 1)ht + a 0 + A ] n ( 2(c 1)ht + a 0 + A ) n A n + A n 1 =(c 1)(h 3 t 2 B n +2ht)+ ( 2(c 1)ht + a 0 + A ) n B n + B n 1 B n =(c 1)(h 3 t 2 +2ht)+ (2(c 1)htB n + a 0 B n + A n )A n + A n 1 B n (2(c 1)htB n + a 0 B n + A n )B n + B n 1 B n B n =(c 1)(h 3 t 2 +2ht)+ (c 1)h2 t + c h 3 t + h The remainder of the proof parallels part (i). (by Lemmas 7 and 9). 4. Fundamental units in real quadratic fields In many cases, it is easy to use the theorems in this paper to write down the fundamental unit in a wide class of real quadratic fields. On [8, p. 119], there is the following statement of the relationship of fundamental units in ( D), where D is a square-free positive rational integer, and the convergents in the continued fraction expansion for D. Theorem 6. Let D be a square-free, positive rational integer, and let K = ( D). Denote by ɛ 0 the fundamental unit of K that exceeds unity, denote by s the period of the continued fraction expansion for D, and denote by P/Q the (s 1)th convergent of it. If D 1 mod 4 or D 1 mod 8, then However, if D 5 mod 8, then ɛ 0 = P + Q D. ɛ 0 = P + Q D. or ɛ 3 0 = P + Q D. Finally, the norm of ɛ 0 is positive if the period s is even, and negative otherwise. We note first of all that the congruence class of f, mod 4, forces c and h to lie in certain congruence classes. We have Table 1. In what follows, let f be a non-square positive integer and let (c, h) be the smallest pair of positive integers satisfying c 2 fh 2 = 1. We denote the Fermat Pell

11 26 j. mclaughlin Table 1. f c h 1 (mod 4) ± 1 (mod 8) 0 (mod 4) 2 (mod 4) ± 1 (mod 16) 0 (mod 4) ± 3 (mod 8) 2 (mod 4) 3 (mod 4) ± 1 (mod 8) 0 (mod 4) 0 (mod 2) 1 (mod 2) polynomials from Theorems 1 5 as follows: f 1 (t) =h 2 t 2 +2ct + f f 2 (t) =(c 1) 2 h 2 t 2 +2(c 1) 2 t + f f 3 (t) =(c +1) 2 h 2 t 2 +2(c +1) 2 t + f (4.1) f 4 (t) =(c +1) 2 h 2 t 2 +2(c 2 1)t + f f 5 (t) =(c 1) 2 h 2 t 2 (h 5 t 2 +4h 2 t +6)+2(c 1)(2c 1)t + f. We consider first the case when f 2 mod 4. From Table 1, c has to be odd and h has to be even, and thus f 2 (t), f 3 (t), f 4 (t) and f 5 (t) are 2 mod 4 for all integral t 0, and f 1 (t) is 2 mod 4 for all even t 0. Thus we can apply Theorem 6. As an illustration, by Theorems 1 and 6, if f(t) =4h 2 t 2 +4ct + f is squarefree, then the fundamental unit in ( f(t)) is 2h 2 t + c + h f(t). With f =22, c = 197 and h = 42, t t 2 is square-free for of the values of t lying between 0 and In particular, it is square-free for t = , so that the fundamental unit in ( ) is immediately known to be Remark 2. The data in this case suggests the following question. Is #{t,t N : t t 2 is square-free} lim = 3 N > N 4? The author is currently unable to provide the answer. A similar situation holds for either of the two possibilities listed in Table 1 for the case when f 3 mod 4. In the first case (c ±1 mod 8 and h 0 mod 4), f 2 (t), f 3 (t), f 4 (t) and f 5 (t) take positive integral values that are 3 mod 4 for all integral t 0, while f 1 (t) takes on positive integral values that are 3 mod 4 for all integral even t 0. In the second case (c 0 mod 2 and h 1 mod 2), all of the polynomials take positive integral values that are 3 mod 4 for all even integral t 0. Theorem 6 can again be used to write down the fundamental unit in infinitely many real quadratic fields. We again consider h 2 t 2 +2ct + f, with t even, and take f = 43, c = 3482 and h = 531. It is found that t t 2 is square-free for of the integers t lying between 0 and In particular, it is squarefree for t = so that the fundamental unit in ( ) is immediately known to be Lastly, we consider the case when f 1 mod 4. For c 1 mod 8, f 2 (t), f 3 (t), f 4 (t) and f 5 (t) f mod 8, for all integral t 0, while f 1 (t) f mod 8 for all integral t 0 mod 4. For c 1 mod 8, f 2 (t), f 3 (t) and f 4 (t) f mod 8 for all integral t 0, f 1 (t) f mod 8 for all positive t 0 mod 4, and f 5 (t) f mod 8 for all even

12 polynomial solutions of pell s equation 27 t 0. For the case when c 0 (mod 2) and h 1 (mod 2), f 1 (t) and f 5 (t) f mod 8 for all positive t 0 mod 4, f 2 (t), f 3 (t) and f 4 (t) f mod 8 for all positive even t. In all of these cases, if f 1 mod 8 and the resulting polynomial is squarefree, then the fundamental unit in the corresponding quadratic field can be written down. This time, we consider the polynomial f 2 (t) =(c 1) 2 h 2 t 2 +2(c 1) 2 t + f. By Theorems 2 and 6, if the continued fraction expansion of f has even period and f 2 (t) is square-free, then the fundamental unit in ( f(t)) is (c 1)h 4 t 2 + 2(c 1)h 2 t + c +(h 3 t + h) f 2 (t). As an illustration, we take f = 57, so that c = 151 and h = 20. f(t) is found to be square-free for of the integers t lying between 0 and In particular, it is square-free for t = , so that the fundamental unit in ( ) is immediately known to be Concluding remarks In this paper, only some limited classes of Fermat Pell polynomials have been considered (for example, in all cases, all but the end and possibly the middle terms in the continued fraction expansion were constant, and the highest degree considered was 4). In a later paper I will consider multivariable Fermat Pell polynomials, polynomials whose continued fraction expansion has all terms non-constant, and single-variable Fermat Pell polynomials where the degree can be arbitrarily large. Some problems still remain for the classes of polynomials examined here. Each of the triple of polynomials examined here constitutes a polynomial solution of Pell s equation by virtue of the fact that c 2 fh 2 = 1 and this alone. A natural question is to determine the continued fraction expansion of f(t) in each of the cases examined when (c, h) (the smallest pair of positive integers satisfying the above integral Pell equation) is replaced by the nth largest pair of such integers. The continued fraction expansion of some of polynomials examined were given in some cases only in special circumstances. For example, the continued fraction expansion of f(t) =(c +1) 2 h 2 t 2 +2(c +1) 2 t + f examined in Theorem 3 was given only in the case where f =[a0 ; a 1,...,a m 1,a m,a m 1,...,a 1, 2a 0 ] where a m = a 0 or a 0 1 and m is even. There remains the problem of determining the continued fraction expansion of f(t) when the length of the period of the continued fraction expansion of f is 0 mod 4 or a m a 0 or a 0 1. Similar problems remain with the continued fraction expansion of the polynomials examined in some of the other theorems. References 1. L. Bernstein, Fundamental units and cycles in the period of real quadratic number fields I, Pacific J. Math. 63 (1976) L. Bernstein, Fundamental units and cycles in the period of real quadratic number fields II, Pacific J. Math. 63 (1976) D. M. Burton, Elementary number theory, 2nd edn (W. C. Brown, Dubuque, IA, 1989). 4. L. Euler, Introduction to analysis of the infinite Book I (Springer, 1988) (first published 1748) (translated by J. D. Blanton). 5. C. Levesque and G. Rhin, A few classes of periodic continued fractions, Utilitas Math. 30 (1986) D. J. Madden, Constructing families of long continued fractions, Pacific J. Math. 198 (2001)

13 28 polynomial solutions of pell s equation 7. R. A. Mollin, Polynomial solutions for the Pell s equation revisited, Indian J. Pure Appl. Math. 28 (1997) W. Narkiewicz, Elementary and analytic theory of algebraic numbers, 2nd edn (Springer/PWN-Polish Scientific Publishers, Warszawa, 1990) (1st edn: 1974). 9. M. B. Nathanson, Polynomial Pell s equations, Proc. Amer. Math. Soc. 56 (1976) O. Perron, Die Lehre von dem Kettenbrüchen (Teubner, Berlin, 1913). 11. A. M. S. Ramasamy, Polynomial solutions for the Pell s equation, Indian J. Pure Appl. Math. 25 (1994) P. Ribenboim, Classical theory of algebraic numbers (Springer, New York, 2001). 13. A. Schinzel, On some problems of the arithmetical theory of continued fractions, Acta Arith. 6 (1960/61) A. Schinzel, On some problems of the arithmetical theory of continued fractions II, Acta Arith. 7 (1961/62) A. J. van der Poorten, Explicit formulas for units in certain quadratic number fields, Algorithmic number theory (Ithaca, NY, 1994), Lecture Notes in Computer Science 877 (Springer, Berlin, 1994) A. J. van der Poorten and H. C. Williams, On certain continued fraction expansions of fixed period length, Acta Arith. 89 (1999) Y. Yamamoto, Real quadratic number fields with large fundamental units, Osaka J. Math. 8 (1971) J. McLaughlin Mathematics Department University of Illinois Champaign Urbana IL USA jgmclaug@math.uiuc.edu

POLYNOMIAL SOLUTIONS TO PELL S EQUATION AND FUNDAMENTAL UNITS IN REAL QUADRATIC FIELDS arxiv: v1 [math.nt] 27 Dec 2018

POLYNOMIAL SOLUTIONS TO PELL S EQUATION AND FUNDAMENTAL UNITS IN REAL QUADRATIC FIELDS arxiv: v1 [math.nt] 27 Dec 2018 POLYNOMIAL SOLUTIONS TO PELL S EQUATION AND FUNDAMENTAL UNITS IN REAL QUADRATIC FIELDS arxiv:1812.10828v1 [math.nt] 27 Dec 2018 J. MC LAUGHLIN Abstract. Finding polynomial solutions to Pell s equation

More information

MULTI-VARIABLE POLYNOMIAL SOLUTIONS TO PELL S EQUATION AND FUNDAMENTAL UNITS IN REAL QUADRATIC FIELDS

MULTI-VARIABLE POLYNOMIAL SOLUTIONS TO PELL S EQUATION AND FUNDAMENTAL UNITS IN REAL QUADRATIC FIELDS MULTI-VARIABLE POLYNOMIAL SOLUTIONS TO PELL S EQUATION AND FUNDAMENTAL UNITS IN REAL QUADRATIC FIELDS J. MC LAUGHLIN Abstract. Solving Pell s equation is of relevance in finding fundamental units in real

More information

R.A. Mollin Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada, T2N 1N4

R.A. Mollin Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada, T2N 1N4 PERIOD LENGTHS OF CONTINUED FRACTIONS INVOLVING FIBONACCI NUMBERS R.A. Mollin Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada, TN 1N e-mail ramollin@math.ucalgary.ca

More information

On the continued fraction expansion of 2 2n+1

On the continued fraction expansion of 2 2n+1 On the continued fraction expansion of n+1 Keith Matthews February 6, 007 Abstract We derive limited information about the period of the continued fraction expansion of n+1 : The period-length is a multiple

More information

Fibonacci Sequence and Continued Fraction Expansions in Real Quadratic Number Fields

Fibonacci Sequence and Continued Fraction Expansions in Real Quadratic Number Fields Malaysian Journal of Mathematical Sciences (): 97-8 (07) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: http://einspem.upm.edu.my/journal Fibonacci Sequence and Continued Fraction Expansions

More information

ON DIFFERENCES OF TWO SQUARES IN SOME QUADRATIC FIELDS

ON DIFFERENCES OF TWO SQUARES IN SOME QUADRATIC FIELDS ON DIFFERENCES OF TWO SQUARES IN SOME QUADRATIC FIELDS ANDREJ DUJELLA AND ZRINKA FRANUŠIĆ Abstract. It this paper, we study the problem of determining the elements in the rings of integers of quadratic

More information

Part II. Number Theory. Year

Part II. Number Theory. Year Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 Paper 3, Section I 1G 70 Explain what is meant by an Euler pseudoprime and a strong pseudoprime. Show that 65 is an Euler

More information

Quadratic Diophantine Equations x 2 Dy 2 = c n

Quadratic Diophantine Equations x 2 Dy 2 = c n Irish Math. Soc. Bulletin 58 2006, 55 68 55 Quadratic Diophantine Equations x 2 Dy 2 c n RICHARD A. MOLLIN Abstract. We consider the Diophantine equation x 2 Dy 2 c n for non-square positive integers D

More information

INDEFINITE QUADRATIC FORMS AND PELL EQUATIONS INVOLVING QUADRATIC IDEALS

INDEFINITE QUADRATIC FORMS AND PELL EQUATIONS INVOLVING QUADRATIC IDEALS INDEFINITE QUADRATIC FORMS AND PELL EQUATIONS INVOLVING QUADRATIC IDEALS AHMET TEKCAN Communicated by Alexandru Zaharescu Let p 1(mod 4) be a prime number, let γ P + p Q be a quadratic irrational, let

More information

A Note on Jacobi Symbols and Continued Fractions

A Note on Jacobi Symbols and Continued Fractions Reprinted from: The American Mathematical Monthly Volume 06, Number, pp 52 56 January, 999 A Note on Jacobi Symbols and Continued Fractions A. J. van der Poorten and P. G. Walsh. INTRODUCTION. It is well

More information

THE DENOMINATORS OF POWER SUMS OF ARITHMETIC PROGRESSIONS. Bernd C. Kellner Göppert Weg 5, Göttingen, Germany

THE DENOMINATORS OF POWER SUMS OF ARITHMETIC PROGRESSIONS. Bernd C. Kellner Göppert Weg 5, Göttingen, Germany #A95 INTEGERS 18 (2018) THE DENOMINATORS OF POWER SUMS OF ARITHMETIC PROGRESSIONS Bernd C. Kellner Göppert Weg 5, 37077 Göttingen, Germany b@bernoulli.org Jonathan Sondow 209 West 97th Street, New Yor,

More information

Newton s formula and continued fraction expansion of d

Newton s formula and continued fraction expansion of d Newton s formula and continued fraction expansion of d ANDREJ DUJELLA Abstract It is known that if the period s(d) of the continued fraction expansion of d satisfies s(d), then all Newton s approximants

More information

On integer solutions to x 2 dy 2 = 1, z 2 2dy 2 = 1

On integer solutions to x 2 dy 2 = 1, z 2 2dy 2 = 1 ACTA ARITHMETICA LXXXII.1 (1997) On integer solutions to x 2 dy 2 = 1, z 2 2dy 2 = 1 by P. G. Walsh (Ottawa, Ont.) 1. Introduction. Let d denote a positive integer. In [7] Ono proves that if the number

More information

Series of Error Terms for Rational Approximations of Irrational Numbers

Series of Error Terms for Rational Approximations of Irrational Numbers 2 3 47 6 23 Journal of Integer Sequences, Vol. 4 20, Article..4 Series of Error Terms for Rational Approximations of Irrational Numbers Carsten Elsner Fachhochschule für die Wirtschaft Hannover Freundallee

More information

SOME RESULTS ON SPECIAL CONTINUED FRACTION EXPANSIONS IN REAL QUADRATIC NUMBER FIELDS

SOME RESULTS ON SPECIAL CONTINUED FRACTION EXPANSIONS IN REAL QUADRATIC NUMBER FIELDS Journal of Mathematical Analysis ISSN: 7-34, URL: http://ilirias.com/jma Volume 7 Issue 4(6), Pages 98-7. SOME RESULTS ON SPECIAL CONTINUED FRACTION EXPANSIONS IN REAL QUADRATIC NUMBER FIELDS ÖZEN ÖZER,

More information

Unisequences and nearest integer continued fraction midpoint criteria for Pell s equation

Unisequences and nearest integer continued fraction midpoint criteria for Pell s equation Unisequences and nearest integer continued fraction midpoint criteria for Pell s equation Keith R. Matthews Department of Mathematics University of Queensland Brisbane Australia 4072 and Centre for Mathematics

More information

Fermat numbers and integers of the form a k + a l + p α

Fermat numbers and integers of the form a k + a l + p α ACTA ARITHMETICA * (200*) Fermat numbers and integers of the form a k + a l + p α by Yong-Gao Chen (Nanjing), Rui Feng (Nanjing) and Nicolas Templier (Montpellier) 1. Introduction. In 1849, A. de Polignac

More information

Explicit solution of a class of quartic Thue equations

Explicit solution of a class of quartic Thue equations ACTA ARITHMETICA LXIV.3 (1993) Explicit solution of a class of quartic Thue equations by Nikos Tzanakis (Iraklion) 1. Introduction. In this paper we deal with the efficient solution of a certain interesting

More information

Math 324 Summer 2012 Elementary Number Theory Notes on Mathematical Induction

Math 324 Summer 2012 Elementary Number Theory Notes on Mathematical Induction Math 4 Summer 01 Elementary Number Theory Notes on Mathematical Induction Principle of Mathematical Induction Recall the following axiom for the set of integers. Well-Ordering Axiom for the Integers If

More information

CLOSED FORM CONTINUED FRACTION EXPANSIONS OF SPECIAL QUADRATIC IRRATIONALS

CLOSED FORM CONTINUED FRACTION EXPANSIONS OF SPECIAL QUADRATIC IRRATIONALS CLOSED FORM CONTINUED FRACTION EXPANSIONS OF SPECIAL QUADRATIC IRRATIONALS DANIEL FISHMAN AND STEVEN J. MILLER ABSTRACT. We derive closed form expressions for the continued fractions of powers of certain

More information

ON THE LENGTH OF THE PERIOD OF A REAL QUADRATIC IRRATIONAL. N. Saradha

ON THE LENGTH OF THE PERIOD OF A REAL QUADRATIC IRRATIONAL. N. Saradha Indian J Pure Appl Math, 48(3): 311-31, September 017 c Indian National Science Academy DOI: 101007/s136-017-09-4 ON THE LENGTH OF THE PERIOD OF A REAL QUADRATIC IRRATIONAL N Saradha School of Mathematics,

More information

On the power-free parts of consecutive integers

On the power-free parts of consecutive integers ACTA ARITHMETICA XC4 (1999) On the power-free parts of consecutive integers by B M M de Weger (Krimpen aan den IJssel) and C E van de Woestijne (Leiden) 1 Introduction and main results Considering the

More information

CALCULATING EXACT CYCLE LENGTHS IN THE GENERALIZED FIBONACCI SEQUENCE MODULO p

CALCULATING EXACT CYCLE LENGTHS IN THE GENERALIZED FIBONACCI SEQUENCE MODULO p CALCULATING EXACT CYCLE LENGTHS IN THE GENERALIZED FIBONACCI SEQUENCE MODULO p DOMINIC VELLA AND ALFRED VELLA. Introduction The cycles that occur in the Fibonacci sequence {F n } n=0 when it is reduced

More information

Equidivisible consecutive integers

Equidivisible consecutive integers & Equidivisible consecutive integers Ivo Düntsch Department of Computer Science Brock University St Catherines, Ontario, L2S 3A1, Canada duentsch@cosc.brocku.ca Roger B. Eggleton Department of Mathematics

More information

MATH 361: NUMBER THEORY FOURTH LECTURE

MATH 361: NUMBER THEORY FOURTH LECTURE MATH 361: NUMBER THEORY FOURTH LECTURE 1. Introduction Everybody knows that three hours after 10:00, the time is 1:00. That is, everybody is familiar with modular arithmetic, the usual arithmetic of the

More information

Necessary and Sufficient Conditions for the Central Norm to Equal 2 h in the Simple Continued Fraction Expansion of 2 h c for Any Odd Non-Square c > 1

Necessary and Sufficient Conditions for the Central Norm to Equal 2 h in the Simple Continued Fraction Expansion of 2 h c for Any Odd Non-Square c > 1 Necessary and Sufficient Conditions for the Central Norm to Equal 2 h in the Simple Continued Fraction Expansion of 2 h c for Any Odd Non-Square c > 1 R.A. Mollin Abstract We look at the simple continued

More information

Abstract. Roma Tre April 18 20, th Mini Symposium of the Roman Number Theory Association (RNTA) Michel Waldschmidt

Abstract. Roma Tre April 18 20, th Mini Symposium of the Roman Number Theory Association (RNTA) Michel Waldschmidt Roma Tre April 8 0, 08 4th Mini Symposium of the Roman umber Theory Association (RTA) Representation of integers by cyclotomic binary forms Michel Waldschmidt Institut de Mathématiques de Jussieu Paris

More information

CONTINUED FRACTIONS Lecture notes, R. M. Dudley, Math Lecture Series, January 15, 2014

CONTINUED FRACTIONS Lecture notes, R. M. Dudley, Math Lecture Series, January 15, 2014 CONTINUED FRACTIONS Lecture notes, R. M. Dudley, Math Lecture Series, January 5, 204. Basic definitions and facts A continued fraction is given by two sequences of numbers {b n } n 0 and {a n } n. One

More information

Notes on Continued Fractions for Math 4400

Notes on Continued Fractions for Math 4400 . Continued fractions. Notes on Continued Fractions for Math 4400 The continued fraction expansion converts a positive real number α into a sequence of natural numbers. Conversely, a sequence of natural

More information

Diophantine quadruples and Fibonacci numbers

Diophantine quadruples and Fibonacci numbers Diophantine quadruples and Fibonacci numbers Andrej Dujella Department of Mathematics, University of Zagreb, Croatia Abstract A Diophantine m-tuple is a set of m positive integers with the property that

More information

Pell Equation x 2 Dy 2 = 2, II

Pell Equation x 2 Dy 2 = 2, II Irish Math Soc Bulletin 54 2004 73 89 73 Pell Equation x 2 Dy 2 2 II AHMET TEKCAN Abstract In this paper solutions of the Pell equation x 2 Dy 2 2 are formulated for a positive non-square integer D using

More information

On Rankin-Cohen Brackets of Eigenforms

On Rankin-Cohen Brackets of Eigenforms On Rankin-Cohen Brackets of Eigenforms Dominic Lanphier and Ramin Takloo-Bighash July 2, 2003 1 Introduction Let f and g be two modular forms of weights k and l on a congruence subgroup Γ. The n th Rankin-Cohen

More information

Another Proof of Nathanson s Theorems

Another Proof of Nathanson s Theorems 1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 14 (2011), Article 11.8.4 Another Proof of Nathanson s Theorems Quan-Hui Yang School of Mathematical Sciences Nanjing Normal University Nanjing 210046

More information

SYMMETRY AND SPECIALIZABILITY IN THE CONTINUED FRACTION EXPANSIONS OF SOME INFINITE PRODUCTS

SYMMETRY AND SPECIALIZABILITY IN THE CONTINUED FRACTION EXPANSIONS OF SOME INFINITE PRODUCTS SYMMETRY AND SPECIALIZABILITY IN THE CONTINUED FRACTION EXPANSIONS OF SOME INFINITE PRODUCTS J MC LAUGHLIN Abstract Let fx Z[x] Set f 0x = x and for n 1 define f nx = ff n 1x We describe several infinite

More information

Irreducible multivariate polynomials obtained from polynomials in fewer variables, II

Irreducible multivariate polynomials obtained from polynomials in fewer variables, II Proc. Indian Acad. Sci. (Math. Sci.) Vol. 121 No. 2 May 2011 pp. 133 141. c Indian Academy of Sciences Irreducible multivariate polynomials obtained from polynomials in fewer variables II NICOLAE CIPRIAN

More information

1. Introduction. Let P and Q be non-zero relatively prime integers, α and β (α > β) be the zeros of x 2 P x + Q, and, for n 0, let

1. Introduction. Let P and Q be non-zero relatively prime integers, α and β (α > β) be the zeros of x 2 P x + Q, and, for n 0, let C O L L O Q U I U M M A T H E M A T I C U M VOL. 78 1998 NO. 1 SQUARES IN LUCAS SEQUENCES HAVING AN EVEN FIRST PARAMETER BY PAULO R I B E N B O I M (KINGSTON, ONTARIO) AND WAYNE L. M c D A N I E L (ST.

More information

On Tornheim s double series

On Tornheim s double series ACTA ARITHMETICA LXXV.2 (1996 On Tornheim s double series by James G. Huard (Buffalo, N.Y., Kenneth S. Williams (Ottawa, Ont. and Zhang Nan-Yue (Beijing 1. Introduction. We call the double infinite series

More information

Carmichael numbers with a totient of the form a 2 + nb 2

Carmichael numbers with a totient of the form a 2 + nb 2 Carmichael numbers with a totient of the form a 2 + nb 2 William D. Banks Department of Mathematics University of Missouri Columbia, MO 65211 USA bankswd@missouri.edu Abstract Let ϕ be the Euler function.

More information

PILLAI S CONJECTURE REVISITED

PILLAI S CONJECTURE REVISITED PILLAI S COJECTURE REVISITED MICHAEL A. BEETT Abstract. We prove a generalization of an old conjecture of Pillai now a theorem of Stroeker and Tijdeman) to the effect that the Diophantine equation 3 x

More information

Beukers integrals and Apéry s recurrences

Beukers integrals and Apéry s recurrences 2 3 47 6 23 Journal of Integer Sequences, Vol. 8 (25), Article 5.. Beukers integrals and Apéry s recurrences Lalit Jain Faculty of Mathematics University of Waterloo Waterloo, Ontario N2L 3G CANADA lkjain@uwaterloo.ca

More information

Summary Slides for MATH 342 June 25, 2018

Summary Slides for MATH 342 June 25, 2018 Summary Slides for MATH 342 June 25, 2018 Summary slides based on Elementary Number Theory and its applications by Kenneth Rosen and The Theory of Numbers by Ivan Niven, Herbert Zuckerman, and Hugh Montgomery.

More information

ON CARMICHAEL NUMBERS IN ARITHMETIC PROGRESSIONS

ON CARMICHAEL NUMBERS IN ARITHMETIC PROGRESSIONS J. Aust. Math. Soc. 88 (2010), 313 321 doi:10.1017/s1446788710000169 ON CARMICHAEL NUMBERS IN ARITHMETIC PROGRESSIONS WILLIAM D. BANKS and CARL POMERANCE (Received 4 September 2009; accepted 4 January

More information

be a sequence of positive integers with a n+1 a n lim inf n > 2. [α a n] α a n

be a sequence of positive integers with a n+1 a n lim inf n > 2. [α a n] α a n Rend. Lincei Mat. Appl. 8 (2007), 295 303 Number theory. A transcendence criterion for infinite products, by PIETRO CORVAJA and JAROSLAV HANČL, communicated on May 2007. ABSTRACT. We prove a transcendence

More information

On Carmichael numbers in arithmetic progressions

On Carmichael numbers in arithmetic progressions On Carmichael numbers in arithmetic progressions William D. Banks Department of Mathematics University of Missouri Columbia, MO 65211 USA bbanks@math.missouri.edu Carl Pomerance Department of Mathematics

More information

PODSYPANIN S PAPER ON THE LENGTH OF THE PERIOD OF A QUADRATIC IRRATIONAL. Copyright 2007

PODSYPANIN S PAPER ON THE LENGTH OF THE PERIOD OF A QUADRATIC IRRATIONAL. Copyright 2007 PODSYPANIN S PAPER ON THE LENGTH OF THE PERIOD OF A QUADRATIC IRRATIONAL JOHN ROBERTSON Copyright 007 1. Introduction The major result in Podsypanin s paper Length of the period of a quadratic irrational

More information

ON THE CONTINUED FRACTION EXPANSION OF THE UNIQUE ROOT IN F(p) OF THE EQUATION x 4 + x 2 T x 1/12 = 0 AND OTHER RELATED HYPERQUADRATIC EXPANSIONS

ON THE CONTINUED FRACTION EXPANSION OF THE UNIQUE ROOT IN F(p) OF THE EQUATION x 4 + x 2 T x 1/12 = 0 AND OTHER RELATED HYPERQUADRATIC EXPANSIONS ON THE CONTINUED FRACTION EXPANSION OF THE UNIQUE ROOT IN F(p) OF THE EQUATION x 4 + x 2 T x 1/12 = 0 AND OTHER RELATED HYPERQUADRATIC EXPANSIONS by A. Lasjaunias Abstract.In 1986, Mills and Robbins observed

More information

On Tornheim's double series

On Tornheim's double series ACTA ARITHMETICA LXXV.2 (1996) On Tornheim's double series JAMES G. HUARD (Buffalo, N.Y.), KENNETH S. WILLIAMS (Ottawa, Ont.) and ZHANG NAN-YUE (Beijing) 1. Introduction. We call the double infinite series

More information

On the existence of unramified p-extensions with prescribed Galois group. Osaka Journal of Mathematics. 47(4) P P.1165

On the existence of unramified p-extensions with prescribed Galois group. Osaka Journal of Mathematics. 47(4) P P.1165 Title Author(s) Citation On the existence of unramified p-extensions with prescribed Galois group Nomura, Akito Osaka Journal of Mathematics. 47(4) P.1159- P.1165 Issue Date 2010-12 Text Version publisher

More information

RANK AND PERIOD OF PRIMES IN THE FIBONACCI SEQUENCE. A TRICHOTOMY

RANK AND PERIOD OF PRIMES IN THE FIBONACCI SEQUENCE. A TRICHOTOMY RANK AND PERIOD OF PRIMES IN THE FIBONACCI SEQUENCE. A TRICHOTOMY Christian Ballot Université de Caen, Caen 14032, France e-mail: ballot@math.unicaen.edu Michele Elia Politecnico di Torino, Torino 10129,

More information

THE PROBLEM OF DIOPHANTUS AND DAVENPORT FOR GAUSSIAN INTEGERS. Andrej Dujella, Zagreb, Croatia

THE PROBLEM OF DIOPHANTUS AND DAVENPORT FOR GAUSSIAN INTEGERS. Andrej Dujella, Zagreb, Croatia THE PROBLEM OF DIOPHANTUS AND DAVENPORT FOR GAUSSIAN INTEGERS Andrej Dujella, Zagreb, Croatia Abstract: A set of Gaussian integers is said to have the property D(z) if the product of its any two distinct

More information

Continued Fractions Expansion of D and Pell Equation x 2 Dy 2 = 1

Continued Fractions Expansion of D and Pell Equation x 2 Dy 2 = 1 Mathematica Moravica Vol. 5-2 (20), 9 27 Continued Fractions Expansion of D and Pell Equation x 2 Dy 2 = Ahmet Tekcan Abstract. Let D be a positive non-square integer. In the first section, we give some

More information

CONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ω(q) AND ν(q)

CONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ω(q) AND ν(q) CONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ωq) AND νq) GEORGE E. ANDREWS, DONNY PASSARY, JAMES A. SELLERS, AND AE JA YEE Abstract. Recently, Andrews, Dixit and Yee introduced partition

More information

PACKING POLYNOMIALS ON SECTORS OF R 2. Caitlin Stanton Department of Mathematics, Stanford University, California

PACKING POLYNOMIALS ON SECTORS OF R 2. Caitlin Stanton Department of Mathematics, Stanford University, California #A67 INTEGERS 14 (014) PACKING POLYNOMIALS ON SECTORS OF R Caitlin Stanton Department of Mathematics, Stanford University, California ckstanton@stanford.edu Received: 10/7/13, Revised: 6/5/14, Accepted:

More information

MATH 4400 SOLUTIONS TO SOME EXERCISES. 1. Chapter 1

MATH 4400 SOLUTIONS TO SOME EXERCISES. 1. Chapter 1 MATH 4400 SOLUTIONS TO SOME EXERCISES 1.1.3. If a b and b c show that a c. 1. Chapter 1 Solution: a b means that b = na and b c that c = mb. Substituting b = na gives c = (mn)a, that is, a c. 1.2.1. Find

More information

Colloq. Math. 145(2016), no. 1, ON SOME UNIVERSAL SUMS OF GENERALIZED POLYGONAL NUMBERS. 1. Introduction. x(x 1) (1.1) p m (x) = (m 2) + x.

Colloq. Math. 145(2016), no. 1, ON SOME UNIVERSAL SUMS OF GENERALIZED POLYGONAL NUMBERS. 1. Introduction. x(x 1) (1.1) p m (x) = (m 2) + x. Colloq. Math. 145(016), no. 1, 149-155. ON SOME UNIVERSAL SUMS OF GENERALIZED POLYGONAL NUMBERS FAN GE AND ZHI-WEI SUN Abstract. For m = 3, 4,... those p m (x) = (m )x(x 1)/ + x with x Z are called generalized

More information

CONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ω(q) AND ν(q)

CONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ω(q) AND ν(q) CONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ωq) AND νq) GEORGE E. ANDREWS, DONNY PASSARY, JAMES A. SELLERS, AND AE JA YEE Abstract. Recently, Andrews, Dixit, and Yee introduced partition

More information

Algebraic integers of small discriminant

Algebraic integers of small discriminant ACTA ARITHMETICA LXXV.4 (1996) Algebraic integers of small discriminant by Jeffrey Lin Thunder and John Wolfskill (DeKalb, Ill.) Introduction. For an algebraic integer α generating a number field K = Q(α),

More information

arxiv: v1 [math.nt] 3 Jan 2019

arxiv: v1 [math.nt] 3 Jan 2019 SOME MORE LONG CONTINUE FRACTIONS, I arxiv:965v mathnt 3 Jan 29 JAMES MC LAUGHLIN AN PETER ZIMMER Abstract In this paper we show how to construct several infinite families of polynomials x,k), such that

More information

1. INTRODUCTION For n G N, where N = {0,1,2,...}, the Bernoulli polynomials, B (t), are defined by means of the generating function

1. INTRODUCTION For n G N, where N = {0,1,2,...}, the Bernoulli polynomials, B (t), are defined by means of the generating function CONGRUENCES RELATING RATIONAL VALUES OF BERNOULLI AND EULER POLYNOMIALS Glenn J. Fox Dept. of Mathematics and Computer Science, Emory University, Atlanta, GA 30322 E-mail: fox@mathcs.emory.edu (Submitted

More information

Journal of Combinatorics and Number Theory 1(2009), no. 1, ON SUMS OF PRIMES AND TRIANGULAR NUMBERS. Zhi-Wei Sun

Journal of Combinatorics and Number Theory 1(2009), no. 1, ON SUMS OF PRIMES AND TRIANGULAR NUMBERS. Zhi-Wei Sun Journal of Combinatorics and Number Theory 1(009), no. 1, 65 76. ON SUMS OF PRIMES AND TRIANGULAR NUMBERS Zhi-Wei Sun Department of Mathematics, Nanjing University Nanjing 10093, People s Republic of China

More information

FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS

FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS Sairaiji, F. Osaka J. Math. 39 (00), 3 43 FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS FUMIO SAIRAIJI (Received March 4, 000) 1. Introduction Let be an elliptic curve over Q. We denote by ˆ

More information

A NOTE ON RATIONAL OPERATOR MONOTONE FUNCTIONS. Masaru Nagisa. Received May 19, 2014 ; revised April 10, (Ax, x) 0 for all x C n.

A NOTE ON RATIONAL OPERATOR MONOTONE FUNCTIONS. Masaru Nagisa. Received May 19, 2014 ; revised April 10, (Ax, x) 0 for all x C n. Scientiae Mathematicae Japonicae Online, e-014, 145 15 145 A NOTE ON RATIONAL OPERATOR MONOTONE FUNCTIONS Masaru Nagisa Received May 19, 014 ; revised April 10, 014 Abstract. Let f be oeprator monotone

More information

ON INTEGERS EXPRESSIBLE BY SOME SPECIAL LINEAR FORM. 1. Introduction

ON INTEGERS EXPRESSIBLE BY SOME SPECIAL LINEAR FORM. 1. Introduction ON INTEGERS EXPRESSIBLE BY SOME SPECIAL LINEAR FORM A. DUBICKAS and A. NOVIKAS Abstract. Let E(4) be the set of positive integers expressible by the form 4M d, where M is a multiple of the product ab and

More information

Section X.55. Cyclotomic Extensions

Section X.55. Cyclotomic Extensions X.55 Cyclotomic Extensions 1 Section X.55. Cyclotomic Extensions Note. In this section we return to a consideration of roots of unity and consider again the cyclic group of roots of unity as encountered

More information

The Maximum Upper Density of a Set of Positive. Real Numbers with no solutions to x + y = kz. John L. Goldwasser. West Virginia University

The Maximum Upper Density of a Set of Positive. Real Numbers with no solutions to x + y = kz. John L. Goldwasser. West Virginia University The Maximum Upper Density of a Set of Positive Real Numbers with no solutions to x + y = z John L. Goldwasser West Virginia University Morgantown, WV 6506 Fan R. K. Chung University ofpennsylvania Philadelphia,

More information

ON MONIC BINARY QUADRATIC FORMS

ON MONIC BINARY QUADRATIC FORMS ON MONIC BINARY QUADRATIC FORMS JEROME T. DIMABAYAO*, VADIM PONOMARENKO**, AND ORLAND JAMES Q. TIGAS*** Abstract. We consider the quadratic form x +mxy +ny, where m n is a prime number. Under the assumption

More information

arxiv:math/ v1 [math.nt] 9 Feb 1998

arxiv:math/ v1 [math.nt] 9 Feb 1998 ECONOMICAL NUMBERS R.G.E. PINCH arxiv:math/9802046v1 [math.nt] 9 Feb 1998 Abstract. A number n is said to be economical if the prime power factorisation of n can be written with no more digits than n itself.

More information

p-regular functions and congruences for Bernoulli and Euler numbers

p-regular functions and congruences for Bernoulli and Euler numbers p-regular functions and congruences for Bernoulli and Euler numbers Zhi-Hong Sun( Huaiyin Normal University Huaian, Jiangsu 223001, PR China http://www.hytc.edu.cn/xsjl/szh Notation: Z the set of integers,

More information

Congruent Number Problem and Elliptic curves

Congruent Number Problem and Elliptic curves Congruent Number Problem and Elliptic curves December 12, 2010 Contents 1 Congruent Number problem 2 1.1 1 is not a congruent number.................................. 2 2 Certain Elliptic Curves 4 3 Using

More information

Notes on Systems of Linear Congruences

Notes on Systems of Linear Congruences MATH 324 Summer 2012 Elementary Number Theory Notes on Systems of Linear Congruences In this note we will discuss systems of linear congruences where the moduli are all different. Definition. Given the

More information

REPRESENTATION OF A POSITIVE INTEGER BY A SUM OF LARGE FOUR SQUARES. Byeong Moon Kim. 1. Introduction

REPRESENTATION OF A POSITIVE INTEGER BY A SUM OF LARGE FOUR SQUARES. Byeong Moon Kim. 1. Introduction Korean J. Math. 24 (2016), No. 1, pp. 71 79 http://dx.doi.org/10.11568/kjm.2016.24.1.71 REPRESENTATION OF A POSITIVE INTEGER BY A SUM OF LARGE FOUR SQUARES Byeong Moon Kim Abstract. In this paper, we determine

More information

A family of quartic Thue inequalities

A family of quartic Thue inequalities A family of quartic Thue inequalities Andrej Dujella and Borka Jadrijević Abstract In this paper we prove that the only primitive solutions of the Thue inequality x 4 4cx 3 y + (6c + 2)x 2 y 2 + 4cxy 3

More information

MATH 25 CLASS 21 NOTES, NOV Contents. 2. Subgroups 2 3. Isomorphisms 4

MATH 25 CLASS 21 NOTES, NOV Contents. 2. Subgroups 2 3. Isomorphisms 4 MATH 25 CLASS 21 NOTES, NOV 7 2011 Contents 1. Groups: definition 1 2. Subgroups 2 3. Isomorphisms 4 1. Groups: definition Even though we have been learning number theory without using any other parts

More information

COMPLEX NUMBERS WITH BOUNDED PARTIAL QUOTIENTS

COMPLEX NUMBERS WITH BOUNDED PARTIAL QUOTIENTS COMPLEX NUMBERS WITH BOUNDED PARTIAL QUOTIENTS WIEB BOSMA AND DAVID GRUENEWALD Abstract. Conjecturally, the only real algebraic numbers with bounded partial quotients in their regular continued fraction

More information

The primitive root theorem

The primitive root theorem The primitive root theorem Mar Steinberger First recall that if R is a ring, then a R is a unit if there exists b R with ab = ba = 1. The collection of all units in R is denoted R and forms a group under

More information

Equivalence of Pepin s and the Lucas-Lehmer Tests

Equivalence of Pepin s and the Lucas-Lehmer Tests EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol., No. 3, 009, (35-360) ISSN 1307-5543 www.ejpam.com Equivalence of Pepin s and the Lucas-Lehmer Tests John H. Jaroma Department of Mathematics & Physics,

More information

On The Weights of Binary Irreducible Cyclic Codes

On The Weights of Binary Irreducible Cyclic Codes On The Weights of Binary Irreducible Cyclic Codes Yves Aubry and Philippe Langevin Université du Sud Toulon-Var, Laboratoire GRIM F-83270 La Garde, France, {langevin,yaubry}@univ-tln.fr, WWW home page:

More information

ON THE LIMIT POINTS OF THE FRACTIONAL PARTS OF POWERS OF PISOT NUMBERS

ON THE LIMIT POINTS OF THE FRACTIONAL PARTS OF POWERS OF PISOT NUMBERS ARCHIVUM MATHEMATICUM (BRNO) Tomus 42 (2006), 151 158 ON THE LIMIT POINTS OF THE FRACTIONAL PARTS OF POWERS OF PISOT NUMBERS ARTŪRAS DUBICKAS Abstract. We consider the sequence of fractional parts {ξα

More information

A PERIODIC APPROACH TO PLANE PARTITION CONGRUENCES

A PERIODIC APPROACH TO PLANE PARTITION CONGRUENCES A PERIODIC APPROACH TO PLANE PARTITION CONGRUENCES MATTHEW S. MIZUHARA, JAMES A. SELLERS, AND HOLLY SWISHER Abstract. Ramanujan s celebrated congruences of the partition function p(n have inspired a vast

More information

Eigenvalues, random walks and Ramanujan graphs

Eigenvalues, random walks and Ramanujan graphs Eigenvalues, random walks and Ramanujan graphs David Ellis 1 The Expander Mixing lemma We have seen that a bounded-degree graph is a good edge-expander if and only if if has large spectral gap If G = (V,

More information

Some distance functions in knot theory

Some distance functions in knot theory Some distance functions in knot theory Jie CHEN Division of Mathematics, Graduate School of Information Sciences, Tohoku University 1 Introduction In this presentation, we focus on three distance functions

More information

THE LIND-LEHMER CONSTANT FOR 3-GROUPS. Stian Clem 1 Cornell University, Ithaca, New York

THE LIND-LEHMER CONSTANT FOR 3-GROUPS. Stian Clem 1 Cornell University, Ithaca, New York #A40 INTEGERS 18 2018) THE LIND-LEHMER CONSTANT FOR 3-GROUPS Stian Clem 1 Cornell University, Ithaca, New York sac369@cornell.edu Christopher Pinner Department of Mathematics, Kansas State University,

More information

THE PROBLEM OF DIOPHANTUS FOR INTEGERS OF. Zrinka Franušić and Ivan Soldo

THE PROBLEM OF DIOPHANTUS FOR INTEGERS OF. Zrinka Franušić and Ivan Soldo THE PROBLEM OF DIOPHANTUS FOR INTEGERS OF Q( ) Zrinka Franušić and Ivan Soldo Abstract. We solve the problem of Diophantus for integers of the quadratic field Q( ) by finding a D()-quadruple in Z[( + )/]

More information

JAMES MC LAUGHLIN S RESEARCH STATEMENT

JAMES MC LAUGHLIN S RESEARCH STATEMENT JAMES MC LAUGHLIN S RESEARCH STATEMENT My research interests are mostly in the field of number theory. Currently, much of my work involves investigations in several areas which have connections to continued

More information

Math 314 Course Notes: Brief description

Math 314 Course Notes: Brief description Brief description These are notes for Math 34, an introductory course in elementary number theory Students are advised to go through all sections in detail and attempt all problems These notes will be

More information

Math 4400/6400 Homework #8 solutions. 1. Let P be an odd integer (not necessarily prime). Show that modulo 2,

Math 4400/6400 Homework #8 solutions. 1. Let P be an odd integer (not necessarily prime). Show that modulo 2, MATH 4400 roblems. Math 4400/6400 Homework # solutions 1. Let P be an odd integer not necessarily rime. Show that modulo, { P 1 0 if P 1, 7 mod, 1 if P 3, mod. Proof. Suose that P 1 mod. Then we can write

More information

SECOND-ORDER RECURRENCES. Lawrence Somer Department of Mathematics, Catholic University of America, Washington, D.C

SECOND-ORDER RECURRENCES. Lawrence Somer Department of Mathematics, Catholic University of America, Washington, D.C p-stability OF DEGENERATE SECOND-ORDER RECURRENCES Lawrence Somer Department of Mathematics, Catholic University of America, Washington, D.C. 20064 Walter Carlip Department of Mathematics and Computer

More information

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2018 57 5. p-adic Numbers 5.1. Motivating examples. We all know that 2 is irrational, so that 2 is not a square in the rational field Q, but that we can

More information

Transformations Preserving the Hankel Transform

Transformations Preserving the Hankel Transform 1 2 3 47 6 23 11 Journal of Integer Sequences, Vol 10 (2007), Article 0773 Transformations Preserving the Hankel Transform Christopher French Department of Mathematics and Statistics Grinnell College Grinnell,

More information

Polygonal Numbers, Primes and Ternary Quadratic Forms

Polygonal Numbers, Primes and Ternary Quadratic Forms Polygonal Numbers, Primes and Ternary Quadratic Forms Zhi-Wei Sun Nanjing University Nanjing 210093, P. R. China zwsun@nju.edu.cn http://math.nju.edu.cn/ zwsun August 26, 2009 Modern number theory has

More information

CONGRUENCES FOR BERNOULLI - LUCAS SUMS

CONGRUENCES FOR BERNOULLI - LUCAS SUMS CONGRUENCES FOR BERNOULLI - LUCAS SUMS PAUL THOMAS YOUNG Abstract. We give strong congruences for sums of the form N BnVn+1 where Bn denotes the Bernoulli number and V n denotes a Lucas sequence of the

More information

#A11 INTEGERS 12 (2012) FIBONACCI VARIATIONS OF A CONJECTURE OF POLIGNAC

#A11 INTEGERS 12 (2012) FIBONACCI VARIATIONS OF A CONJECTURE OF POLIGNAC #A11 INTEGERS 12 (2012) FIBONACCI VARIATIONS OF A CONJECTURE OF POLIGNAC Lenny Jones Department of Mathematics, Shippensburg University, Shippensburg, Pennsylvania lkjone@ship.edu Received: 9/17/10, Revised:

More information

CS 5319 Advanced Discrete Structure. Lecture 9: Introduction to Number Theory II

CS 5319 Advanced Discrete Structure. Lecture 9: Introduction to Number Theory II CS 5319 Advanced Discrete Structure Lecture 9: Introduction to Number Theory II Divisibility Outline Greatest Common Divisor Fundamental Theorem of Arithmetic Modular Arithmetic Euler Phi Function RSA

More information

Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi

Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.

More information

SQUARE PATTERNS AND INFINITUDE OF PRIMES

SQUARE PATTERNS AND INFINITUDE OF PRIMES SQUARE PATTERNS AND INFINITUDE OF PRIMES KEITH CONRAD 1. Introduction Numerical data suggest the following patterns for prime numbers p: 1 mod p p = 2 or p 1 mod 4, 2 mod p p = 2 or p 1, 7 mod 8, 2 mod

More information

The Continuing Story of Zeta

The Continuing Story of Zeta The Continuing Story of Zeta Graham Everest, Christian Röttger and Tom Ward November 3, 2006. EULER S GHOST. We can only guess at the number of careers in mathematics which have been launched by the sheer

More information

Pell s Equation Claire Larkin

Pell s Equation Claire Larkin Pell s Equation is a Diophantine equation in the form: Pell s Equation Claire Larkin The Equation x 2 dy 2 = where x and y are both integer solutions and n is a positive nonsquare integer. A diophantine

More information

A new proof of Euler s theorem. on Catalan s equation

A new proof of Euler s theorem. on Catalan s equation Journal of Applied Mathematics & Bioinformatics, vol.5, no.4, 2015, 99-106 ISSN: 1792-6602 (print), 1792-6939 (online) Scienpress Ltd, 2015 A new proof of Euler s theorem on Catalan s equation Olufemi

More information

Newton, Fermat, and Exactly Realizable Sequences

Newton, Fermat, and Exactly Realizable Sequences 1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 8 (2005), Article 05.1.2 Newton, Fermat, and Exactly Realizable Sequences Bau-Sen Du Institute of Mathematics Academia Sinica Taipei 115 TAIWAN mabsdu@sinica.edu.tw

More information

DIVISIBILITY PROPERTIES OF THE 5-REGULAR AND 13-REGULAR PARTITION FUNCTIONS

DIVISIBILITY PROPERTIES OF THE 5-REGULAR AND 13-REGULAR PARTITION FUNCTIONS INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (008), #A60 DIVISIBILITY PROPERTIES OF THE 5-REGULAR AND 13-REGULAR PARTITION FUNCTIONS Neil Calkin Department of Mathematical Sciences, Clemson

More information