Piatetski-Shapiro primes from almost primes
|
|
- Brett Stewart
- 5 years ago
- Views:
Transcription
1 Piatetski-Shapiro primes from almost primes Roger C. Baker Department of Mathematics, Brigham Young University Provo, UT USA William D. Banks Department of Mathematics, University of Missouri Columbia, MO USA Zhenyu V. Guo Department of Mathematics, University of Missouri Columbia, MO USA Aaron M. Yeager Department of Mathematics, University of Missouri Columbia, MO USA Abstract Let be the floor function. In this paper, we show that for any fixed c (1, ) there are infinitely many primes of the form p = nc, where n is a natural number with at most eight prime factors (counted with multiplicity). 1
2 1 Introduction The Piatetski-Shapiro sequences are those sequences of the form ( n c ) n N (c > 1, c N), where t denotes the integer part of any t R. Such sequences are named in honor of Piatetski-Shapiro, who showed in [6] that for any number c (1, 12) 11 the set P (c) := { p prime : p = n c for some n N } is infinite. The admissible range for c in this result has been extended many times over the years and is currently known for all c (1, 243 ) thanks to 205 Rivat and Wu [8]. For any natural number r, let N r denote the set of r-almost primes, i.e., the set of natural numbers having at most r prime factors, counted with multiplicity. In this paper, we introduce and study sets of Piatetski-Shapiro primes of the form P (c) r := { p prime : p = n c for some n N r }. Our main result is the following: Theorem 1. For any fixed c (1, 77 ) the set P(c) 76 8 is infinite. More precisely, { n x : n N 8 and n c is prime } x (log x) 2, where the implied constant in the symbol depends only on c. 2 Notation Throughout the paper, we set γ := 1/c for a given real number c > 1. The parameters ε, δ are positive real numbers that are sufficiently small for all of our purposes. Any implied constants in the symbols O,,, may depend on c, ε, δ but are absolute otherwise. The letter p always denotes a prime number, and Λ is used to denote the von Mangoldt function. We use notation of the form m M as an abbreviation for M < m 2M. 2
3 As is customary, we put e(t) := e 2πit and {t} := t t (t R). Throughout the paper, we make considerable use of the sawtooth function defined by ψ(t) := t t 1 2 = {t} 1 2 (t R) (1) as well as the well known approximation of Vaaler [10]: for any H 1 there exist numbers a h and b h such that ψ(t) a h e(th) b h e(th), a h 1 h, b h 1 H. (2) 0< h H h H 3 Proof of Theorem Initial approach We analyze exponential sums that are relevant for finding primes in P (c) r the number r as small as possible. The set that we sieve is A := { n x : n c is prime }. with For any d D, where D is a fixed power of x to be specified later, we must estimate accurately the cardinality of Since md A if and only if A d := { n A : d n }. p (md) c < p + 1 and md x, to within O(1) the cardinality of A d is equal to the number of primes p x c for which the interval [ p γ d 1, (p + 1) γ d 1) contains a natural number; thus, A d = { p x c : (p + 1) γ d 1 < m p γ d 1 for some m N } + O(1) = p x c ( p γ d 1 (p + 1) γ d 1 ) + O(1) = X d 1 + p x c ( ψ( (p + 1) γ d 1 ) ψ( p γ d 1 ) ) + O(1), 3
4 where ψ is given by (1), and X := p x c ((p + 1) γ p γ ) = p x c γp γ 1 + O(1) x c log x (x ). It is unnecessary to evaluate X more precisely than this; however, for any sufficiently small ε > 0 we need to show that Ad X d 1 X x ε/3 x1 ε/3 log x d D (x ). (3) Splitting the range of d into dyadic subintervals and using partial summation in a standard way, it suffices to prove that the bound Λ(n) ( ψ( (n + 1) γ d 1 ) ψ( n γ d 1 ) ) x1 ε/2 (4) d D 1 holds uniformly for D 1 D, N x c, N 1 N. In turn, (4) is an immediate consequence of the uniform bound Λ(n) ( ψ( (n + 1) γ d 1 ) ψ( n γ d 1 ) ) x 1 ε/2 d 1 (5) for d D, N x c, N 1 N. Our aim is to establish (5) with D as large as possible, and in this subsection we show that (5) holds when D x 1 136c/157 (6) and ε > 0 is sufficiently small. Suppose this has been done, and observe that 1 136c 157 > 8 63 whenever c < Then, for any fixed c (1, 8635) and α ( 8, 1 136c ), the inequality (3) with D := x α implies the bound A d X d 1 d D X (log X ) 2, thus we can apply the weighted sieve in the form [4, Ch. 5, Prop. 1] with the choices R := 8, δ R := and g :=
5 Note that g < R δ R, and (if x is large enough) the inequality a < D g holds for all a A since αg > 1; thus, the conditions of [4, Ch. 5, Prop. 1] are met, and we conclude that A contains at least X / log X numbers with at most eight prime factors. This yields the statement of the theorem for all c in the interval (1, 8635) We now turn to the proof of (5) for all D satisfying (6). Let S denote the sum on the left side of (5). From Vaaler s approximation (2) we derive the inequality S S 1 ( b h e( h(n + 1) γ d 1 ) + ) b h e( hn γ d 1 ), where S 1 := Λ(n) Λ(n) h H 0< h H h H a h ( e( h(n + 1) γ d 1 ) e( hn γ d 1 ) ). Because h H b h e(th) is nonnegative for all t R, it follows that S S 1 log N 1 = log N 1 h H ( h H b h e( h(n + 1) γ d 1 ) + h H b h e( hn γ d 1 ) b h ( S(h, 0) + S(h, 1) ), (7) ) where we have put S(h, j) := e( h(n + j) γ d 1 ). We now choose H := x 1+ε Nd (8) so that the contribution from h = 0 on the right side of (7) does not exceed 2b 0 N log N 1 x 1 ε/2 d 1. On the other hand, using the exponent pair ( 1, 1 ) we derive that 2 2 S(h, j) h 1/2 N γ/2 d 1/2 + h 1 N 1 γ d (0 < h H, j = 0, 1). 5
6 We bound the contribution in (7) from integers h in the range 0 < h H by observing that b h h 1/2 N γ/2 d 1/2 H 1/2 N γ/2 d 1/2 = x 1/2+ε/2 N 1/2+γ/2 x 1 ε d 1 0< h H since and that since d D x 1 c/2 2ε x 3/2 3ε/2 N 1/2 γ/2, 0< h H b h h 1 N 1 γ d N 1 γ d x 1 ε d 1 d 2 D 2 x 2 c ε x 1 ε N γ 1. Putting everything together, we conclude that S S 1 + O(x 1 ε/2 d 1 ), and it remains only to bound S 1. Next, we use a partial summation argument from the book of Graham and Kolesnik [3]. Writing S 1 = a h Λ(n)φ h (n)e(hn γ d 1 ) with 0< h H 0< h H φ h (t) := e(h(t + 1) γ d 1 ht γ d 1 ) 1, we would like to show that h 1 Λ(n)φ h (n) e(hn γ d 1 ) x1 ε d 1 (d D). (9) Taking into account the bounds φ h (t) hn γ 1 d 1 and φ h(t) hn γ 2 d 1 (N t N 1 ), 6
7 the left side of (9) is, on integrating by parts, bounded by h 1 φ h (N 1 ) Λ(n) e(hn γ d 1 ) h H + N1 h 1 φ h(t) Λ(n) e(hn γ d 1 ) dt h H N N<n t N γ 1 d 1 Λ(n) e(hn γ d 1 ) h H N<n N 2 for some number N 2 (N, N 1 ]. Therefore, it suffices to show that the bound Λ(n) e(hn γ d 1 ) x1 ε N 1 γ (10) N<n N 2 h H holds uniformly for d D, N x c, N 2 N. To establish (10) we use the decomposition of Heath-Brown [5]; it suffices to show that our type I and type II sums satisfy the uniform bounds S I := a l e(hl γ m γ d 1 ) x1 2ε N 1 γ, (11) S II := H 1 h H 2 H 1 h H 2 l L l L m M lm J a l b m e(hl γ m γ d 1 ) x1 2ε N 1 γ, (12) m M lm J in some specific ranges. Here, J is an interval in (N, N 1 ], H 1 H, H 2 H 1, LM N, and the numbers a l, b m C satisfy a l 1, b m 1. In view of [5, pp ] we need to show that (12) holds uniformly for all L in the range u L N 1/3 for some u x ε N 1/5, (13) and for such u we need to show that (11) holds uniformly for all M satisfying M N 1/2 u 1/2. Put F := H 1 N γ d 1. For the type II sum, we apply Baker [1, Thm. 2], which yields the bound S II ( T II,1 + T II,2 ) H1 Nx ε 7
8 with T II,1 := (H 1 L) 1/2 and T II,2 := ( ) k/(2+2k) F M (1+k l)/(2+2k) H 1 L for any exponent pair (k, l) provided that F H 1 L. For the type I sum, by Robert and Sargos [9, Thm. 3] we have the bound with T I,1 := S I ( T I,1 + T I,2 + T I,3 ) H1 Nx ε ( ) 1/4 F, T H 1 LM 2 I,2 := M 1/2 and T I,3 := F 1. Hence, to establish (11) and (12) it suffices to verify that max { T I,1, T I,2, T I,3, T II,1, T II,2 } x 1 3ε H 1 1 N γ. (14) From the definition of F we see that the bound T I,3 = F 1 x 1 3ε H 1 1 N γ (15) is equivalent to d x 1 3ε and thus follows from the inequality D x 1 3ε which is implied by (6). To guarantee that holds for all L u we simply define T II,1 = (H 1 L) 1/2 x 1 3ε H 1 1 N γ (16) u := x 2+6ε H 1 N 2γ. We need to check that the condition u x ε N 1/5 of (13) is met. For this, taking into account (8), it suffices to have D x 3 8ε N 4/5 2γ. The worst case occurs when N = x c, leading to the constraint D x 1 4c/5 8ε, which follows from (6). Next, if M satisfies the lower bound M N 1/2 u 1/2 = x 1 3ε H 1/2 1 N 1/2 γ, 8
9 then the upper bound holds; therefore, the bound holds provided that M 1/2 x 1/2+2ε H 1/4 1 N 1/4+γ/2 (17) T I,2 = M 1/2 x 1 3ε H 1 1 N γ (18) H 1 x 6/5 4ε N 1/5 6γ/5. Using (8) again, we see that (18) follows from the inequality D x 11/5 5ε N 4/5 6γ/5. Taking N := x c leads to the restriction D x 1 4c/5 5ε, and this is implied by (6). Next, using the definition of F and the relation LM N one sees that the bound ( ) 1/4 F T I,1 = x 1 3ε H 1 H 1 LM 2 1 N γ (19) holds whenever H 1 M 1/4 d 1/4 x 1 3ε N 1/4 5γ/4. Taking into account (8) and (17) this bound follows from the condition D x 19/7 7ε N 6/7 12γ/7. With N := x c we derive the constraint D x 1 6c/7 7ε, which is a consequence of (6). Our next goal is to establish the bound ( ) N γ k/(2+2k) T II,2 = M (1+k l)/(2+2k) x 1 3ε H1 1 N γ. (20) Ld To begin, we check that the condition F H 1 L is met, or equivalently, that d N γ L 1. Since L N 1/3 for the type II sum, it is enough to have D N γ 1/3 ε. (21) But this follows essentially from (6). Indeed, since c > 1 and γ := 1/c, the inequality (1 136c/157)(2/3 + 2γ) < 2(γ 1/3) 9
10 is easily verified, and we have which implies that D 2/3+2γ ε ( x 1 136c/157) 2/3+2γ ε ( x 2 3ε ) γ 1/3 ε, ( ) x D 1+γ 2 3ε γ 1/3 ε. D On the other hand, we can certainly assume that HN x 1 2ε N 1 γ, for otherwise (11) and (12) are trivial; therefore N 1+γ x 2 3ε d 1, and we have ( ) x D 1+γ 2 3ε γ 1/3 ε ( ) x 2 3ε γ 1/3 ε ( N 1+γ) γ 1/3 ɛ, D d which yields (21). Using the relation LM N, the lower bound M N 2/3 (which follows from L N 1/3 ) and the definition (8), we see that the bound (20) holds if d ν k x 2ν 4νε N ν γν γk+k+2(1 l)/3, where we have put ν := 2k + 2. The exponent of N is negative since k + 2(1 l)/3 < k + 1/3 < ν/2; therefore, the worst case occurs when N = x c, and it suffices to have where µ := D x 1 cµ/3 2ε, (22) ν k 2(1 l)/3 ν k = 3k + 2l + 4 k + 2 With the choice (k, l) := ( 57, 64 ) (which is BA5 ( 1, 1 ) in the notation of 2 2 Graham [2]) we have µ 3 = < , and therefore (22) follows from (6). This proves (20). Combining the bounds (15), (16), (18), (19) and (20), we obtain (14), and this completes the proof of Theorem 1 for c (1, 8635)
11 3.2 Refinement Here, we extend the ideas of 3.1 to show that for any δ > 0, the bound (5) holds for all sufficiently small ε > 0 (depending on δ) under the less stringent condition that 380c 1 D x 441 δ. (23) After this has been done, taking into account that 1 380c 441 > 8 63 whenever c < 77 76, the proof of Theorem 1 for the full range c (1, 77 ) is completed using the 76 sieve argument presented after (6). Following Rivat and Sargos [7, Lem. 2] it suffices to show that (i) The type II bound (12) holds for L in the range u 0 L u 2 0 for some u 0 [N 1/10, N 1/6 ]; (ii) For such u 0, the type I bound (11) holds whenever M N 1/2 u 1/2 0 ; (iii) For such u 0 and any numbers a m, c h C with a m 1, c h 1, the type I bound S I := c h a m e(hl γ m γ d 1 ) x 1 2ε N 1 γ h H m M l L holds whenever u 2 0 L N 1/3. Taking u := x 2+6ε H 1 N 2γ as in 3.1, we put u 0 := max{n 1/10, u}. The condition u 0 N 1/6 follows from the inequality x 2+6ε HN 2γ N 1/6, which in view of (8) is implied by D x 3 7ε N 5/6 2γ. Taking N := x c leads to D x 1 5c/6 7ε, which follows from (23). To verify condition (i) we again use the bound S II ( T II,1 + T II,2 ) H1 Nx ε, 11
12 but we now make the simple choice (k, l) := ( 1, 1 ), so that 2 2 ( ) N γ 1/6 T II,2 = M 1/3. Ld Since u 0 u the bound (16) for T II,1 remains valid in this case. As for T II,2 we have T II,2 N (γ 1)/6 d 1/6 M 1/6, and we now suppose that M Nu 2 0 = min { } N 4/5, N 1 4γ x 4 12ε H1 2. Thus, to obtain (12) we need both of the inequalities to hold: N (γ 1)/6 d 1/6 (N 4/5 ) 1/6 x 1 3ε H1 1 N γ, (24) N (γ 1)/6 d 1/6 (N 1 4γ x 4 12ε H1 2 ) 1/6 x 1 3ε H1 1 N γ. (25) Since H 1 x 1+ε Nd and d D, the first inequality (24) is a consequence of the bound D x 12/5 5ε N 21/25 7γ/5, which is satisfied since N x c and D x 1 21c/25 5ε. Similarly, the inequality (25) follows from D x 18/7 6ε N 6/7 11γ/7, which is satisfied since N x c and D x 1 6c/7 6ε. Condition (ii) also follows from 3.1 in the case that u N 1/10. When u < N 1/10 it suffices to show that max { T I,1, T I,2, T I,3 } x 1 3ε H 1 1 N γ (26) when M N 9/20. Taking into account (8) we see that the bound T I,2 = M 1/2 x 1 3ε H 1 1 N γ (27) holds provided that D x 2 4ε N 31/40 γ. The worst case N = x c leads to D x 1 31c/40 4ε, and since 380 implied by (23). We also know that (19) holds whenever > this is H 1 M 1/4 d 1/4 x 1 3ε N 1/4 5γ/4. 12
13 Taking into account (8) this bound follows from the inequality D x 8/3 3ε N 51/60 5γ/3. With N := x c we derive the constraint D x 1 51c/60 3ε, and as 380 > this a consequence of (23). Combining (15), (19) and (27) we obtain (26) as required. It remains to verify condition (iii). Rather than adapting Rivat and Sargos [7], we quote an abstraction of their method due to Wu [11]. Taking k := 5 in [11, Thm. 2] we have (in Wu s notation) a bound of the form (log x) 1 S I (X 32 H 114 M 147 N 137 ) 1/ However, in place of (X, H, M, N) we use the quadruple (H 1 N γ d 1, M, H 1, L). The triple of exponents (α, β, γ) becomes (γ, 1, γ) in our case, and it is straightforward to check that the various hypotheses of [11, Thm. 2] are satisfied. Applying the theorem, it follows that where S I ( T I,1 + T I,2 + T I,3 + T I,4 + T I,5 + T I,6 + T I,7) x ε, T I,1 := ( H N γ L 23 d 32) 1/174, TI,5 := H 1/2 1 NL 1/2, T I,2 := H 3/4 1 N 1/2+γ/4 L 1/2 d 1/4, T I,6 := H 1 N 1/2 L 1/2, T I,3 := H 5/4 1 N 1/2+γ/4 d 1/4, T I,7 := H 1/2 1 N 1 γ/2 d 1/2. T I,4 := H 1 NL 1, It suffices to show that given that max { T I,1, T I,2, T I,3, T I,4, T I,5, T I,6, T I,7} x 1 3ε N 1 γ (28) N x c, H 1 x 1+ε Nd and N 1/5 L N 1/3. (29) First, we note that the bound T I,1 = ( H N γ L 23 d 32) 1/174 x 1 3ε N 1 γ (30) is equivalent to H N γ L 23 d 32 x ε. 13
14 Using the first two inequalities and the upper bound on L in (29), it suffices to have D 147 x c/3 701ε, which is (23). Similarly, the bounds follow from the inequalities T I,2 = H 3/4 1 N 1/2+γ/4 L 1/2 d 1/4 x 1 3ε N 1 γ, (31) T I,6 = H 1 N 1/2 L 1/2 x 1 3ε N 1 γ, (32) D x 1 5c/6 15ε/2 and D x 1 2c/3 4ε, respectively, and these are easy consequences of (23) since > 5 6 > 2 3. On the other hand, using the first two inequalities and the lower bound on L in (29), we see that both bounds follow from the inequality T I,4 = H 1 NL 1 x 1 3ε N 1 γ, (33) T I,5 = H 1/2 1 NL 1/2 x 1 3ε N 1 γ, (34) D x 1 4c/5 7ε, which is implied by (23) since 380 > 4. Next, using the first two inequalities in (29) and disregarding the bounds on L, it is easy to check that follows from T I,3 = H 5/4 1 N 1/2+γ/4 d 1/4 x 1 3ε N 1 γ (35) D x 1 3c/4 17ε/4, which is implied by (23) since > 3 4. Similarly, is a consequence of the inequality T I,7 = H 1/2 1 N 1 γ/2 d 1/2 x 1 3ε N 1 γ (36) D x 1 c/2 7ε/2, which follows from (23) since > 1 2. Combining the bounds (30), (31), (32), (33), (34), (35) and (36), we obtain (28), and Theorem 1 is proved. 14
15 References [1] R. C. Baker, The square-free divisor problem, Quart. J. Math. Oxford 45 (1994), [2] S. W. Graham, An algorithm for computing optimal exponent pairs, J. London Math. Soc. (2) 33 (1986), no. 2, [3] S. W. Graham and G. Kolesnik, Van der Corput s method of exponential sums. London Mathematical Society Lecture Note Series, 126. Cambridge University Press, Cambridge, [4] G. Greaves, Sieves in Number Theory. Results in Mathematics and Related Areas (3), 43. Springer-Verlag, Berlin, [5] D. R. Heath-Brown, Prime numbers in short intervals and a generalized Vaughan identity, Canad. J. Math. 34 (1982), no. 6, [6] I. I. Piatetski-Shapiro, On the distribution of prime numbers in the sequence of the form f(n), Mat. Sb. 33 (1953), [7] J. Rivat and P. Sargos, Nombres premiers de la forme n c, Canad. J. Math. 53 (2001), no. 2, [8] J. Rivat and J. Wu, Prime numbers of the form n c, Glasg. Math. J. 43 (2001), no. 2, [9] O. Robert and P. Sargos, Three-dimensional exponential sums with monomials, J. Reine Angew. Math. 591 (2006), [10] J. D. Vaaler, Some extremal problems in Fourier analysis, Bull. Amer. Math. Soc. 12 (1985), [11] J. Wu, On the primitive circle problem, Monatsh. Math. 135 (2002), no. 1,
Almost Primes of the Form p c
Almost Primes of the Form p c University of Missouri zgbmf@mail.missouri.edu Pre-Conference Workshop of Elementary, Analytic, and Algorithmic Number Theory Conference in Honor of Carl Pomerance s 70th
More informationA HYBRID OF TWO THEOREMS OF PIATETSKI-SHAPIRO
A HYBRID OF TWO THEOREMS OF PIATETSKI-SHAPIRO ANGEL KUMCHEV AND ZHIVKO PETROV Abstract. Let c > 1 and 0 < γ < 1 be real, with c / N. We study the solubility of the Diophantine inequality p c 1 + p c 2
More informationDensity of non-residues in Burgess-type intervals and applications
Bull. London Math. Soc. 40 2008) 88 96 C 2008 London Mathematical Society doi:0.2/blms/bdm Density of non-residues in Burgess-type intervals and applications W. D. Banks, M. Z. Garaev, D. R. Heath-Brown
More informationClusters of primes with square-free translates
Submitted to Rev. Mat. Iberoam., 1 22 c European Mathematical Society Clusters of primes with square-free translates Roger C. Baker and Paul Pollack Abstract. Let R be a finite set of integers satisfying
More informationOn Gauss sums and the evaluation of Stechkin s constant
On Gauss sums and the evaluation of Stechkin s constant William D. Banks Department of Mathematics University of Missouri Columbia, MO 65211 USA bankswd@missouri.edu Igor E. Shparlinski Department of Pure
More informationLes chiffres des nombres premiers. (Digits of prime numbers)
Les chiffres des nombres premiers (Digits of prime numbers) Joël RIVAT Institut de Mathématiques de Marseille, UMR 7373, Université d Aix-Marseille, France. joel.rivat@univ-amu.fr soutenu par le projet
More informationCarmichael 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 informationCharacter sums with Beatty sequences on Burgess-type intervals
Character sums with Beatty sequences on Burgess-type intervals William D. Banks Department of Mathematics University of Missouri Columbia, MO 65211 USA bbanks@math.missouri.edu Igor E. Shparlinski Department
More informationExponential and character sums with Mersenne numbers
Exponential and character sums with Mersenne numbers William D. Banks Dept. of Mathematics, University of Missouri Columbia, MO 652, USA bankswd@missouri.edu John B. Friedlander Dept. of Mathematics, University
More informationA Diophantine Inequality Involving Prime Powers
A Diophantine Inequality Involving Prime Powers A. Kumchev 1 Introduction In 1952 I. I. Piatetski-Shapiro [8] studied the inequality (1.1) p c 1 + p c 2 + + p c s N < ε where c > 1 is not an integer, ε
More informationOn the distribution of consecutive square-free
On the distribution of consecutive square-free numbers of the form [n],[n]+ S. I. Dimitrov Faculty of Applied Mathematics and Informatics, Technical University of Sofia 8, St.Kliment Ohridsi Blvd. 756
More informationPrime divisors in Beatty sequences
Journal of Number Theory 123 (2007) 413 425 www.elsevier.com/locate/jnt Prime divisors in Beatty sequences William D. Banks a,, Igor E. Shparlinski b a Department of Mathematics, University of Missouri,
More informationOptimal primitive sets with restricted primes
Optimal primitive sets with restricted primes arxiv:30.0948v [math.nt] 5 Jan 203 William D. Banks Department of Mathematics University of Missouri Columbia, MO 652 USA bankswd@missouri.edu Greg Martin
More information#A69 INTEGERS 13 (2013) OPTIMAL PRIMITIVE SETS WITH RESTRICTED PRIMES
#A69 INTEGERS 3 (203) OPTIMAL PRIMITIVE SETS WITH RESTRICTED PRIMES William D. Banks Department of Mathematics, University of Missouri, Columbia, Missouri bankswd@missouri.edu Greg Martin Department of
More informationERIC LARSON AND LARRY ROLEN
PROGRESS TOWARDS COUNTING D 5 QUINTIC FIELDS ERIC LARSON AND LARRY ROLEN Abstract. Let N5, D 5, X) be the number of quintic number fields whose Galois closure has Galois group D 5 and whose discriminant
More informationOn 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 informationA combinatorial problem related to Mahler s measure
A combinatorial problem related to Mahler s measure W. Duke ABSTRACT. We give a generalization of a result of Myerson on the asymptotic behavior of norms of certain Gaussian periods. The proof exploits
More information1 i<j k (g ih j g j h i ) 0.
CONSECUTIVE PRIMES IN TUPLES WILLIAM D. BANKS, TRISTAN FREIBERG, AND CAROLINE L. TURNAGE-BUTTERBAUGH Abstract. In a stunning new advance towards the Prime k-tuple Conjecture, Maynard and Tao have shown
More informationOn a diophantine inequality involving prime numbers
ACTA ARITHMETICA LXI.3 (992 On a diophantine inequality involving prime numbers by D. I. Tolev (Plovdiv In 952 Piatetski-Shapiro [4] considered the following analogue of the Goldbach Waring problem. Assume
More informationInjective semigroup-algebras
Injective semigroup-algebras J. J. Green June 5, 2002 Abstract Semigroups S for which the Banach algebra l (S) is injective are investigated and an application to the work of O. Yu. Aristov is described.
More informationON 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 informationIntegers without large prime factors in short intervals and arithmetic progressions
ACTA ARITHMETICA XCI.3 (1999 Integers without large prime factors in short intervals and arithmetic progressions by Glyn Harman (Cardiff 1. Introduction. Let Ψ(x, u denote the number of integers up to
More informationOn the digits of prime numbers
On the digits of prime numbers Joël RIVAT Institut de Mathématiques de Luminy, Université d Aix-Marseille, France. rivat@iml.univ-mrs.fr work in collaboration with Christian MAUDUIT (Marseille) 1 p is
More informationSome Arithmetic Functions Involving Exponential Divisors
2 3 47 6 23 Journal of Integer Sequences, Vol. 3 200, Article 0.3.7 Some Arithmetic Functions Involving Exponential Divisors Xiaodong Cao Department of Mathematics and Physics Beijing Institute of Petro-Chemical
More informationSYMMETRIC INTEGRALS DO NOT HAVE THE MARCINKIEWICZ PROPERTY
RESEARCH Real Analysis Exchange Vol. 21(2), 1995 96, pp. 510 520 V. A. Skvortsov, Department of Mathematics, Moscow State University, Moscow 119899, Russia B. S. Thomson, Department of Mathematics, Simon
More informationCongruences involving product of intervals and sets with small multiplicative doubling modulo a prime
Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime J. Cilleruelo and M. Z. Garaev Abstract We obtain a sharp upper bound estimate of the form Hp o(1)
More informationOn Gelfond s conjecture on the sum-of-digits function
On Gelfond s conjecture on the sum-of-digits function Joël RIVAT work in collaboration with Christian MAUDUIT Institut de Mathématiques de Luminy CNRS-UMR 6206, Aix-Marseille Université, France. rivat@iml.univ-mrs.fr
More informationThe density of rational points on non-singular hypersurfaces, I
The density of rational points on non-singular hypersurfaces, I T.D. Browning 1 and D.R. Heath-Brown 2 1 School of Mathematics, Bristol University, Bristol BS8 1TW 2 Mathematical Institute,24 29 St. Giles,Oxford
More informationPrime Divisors of Palindromes
Prime Divisors of Palindromes William D. Banks Department of Mathematics, University of Missouri Columbia, MO 6511 USA bbanks@math.missouri.edu Igor E. Shparlinski Department of Computing, Macquarie University
More informationPrime Number Theory and the Riemann Zeta-Function
5262589 - Recent Perspectives in Random Matrix Theory and Number Theory Prime Number Theory and the Riemann Zeta-Function D.R. Heath-Brown Primes An integer p N is said to be prime if p and there is no
More informationActa Mathematica Academiae Paedagogicae Nyíregyháziensis 32 (2016), ISSN
Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 3 06, 3 3 www.emis.de/journals ISSN 786-009 A NOTE OF THREE PRIME REPRESENTATION PROBLEMS SHICHUN YANG AND ALAIN TOGBÉ Abstract. In this note, we
More informationDistribution of Fourier coefficients of primitive forms
Distribution of Fourier coefficients of primitive forms Jie WU Institut Élie Cartan Nancy CNRS et Nancy-Université, France Clermont-Ferrand, le 25 Juin 2008 2 Presented work [1] E. Kowalski, O. Robert
More informationPRIME-REPRESENTING FUNCTIONS
Acta Math. Hungar., 128 (4) (2010), 307 314. DOI: 10.1007/s10474-010-9191-x First published online March 18, 2010 PRIME-REPRESENTING FUNCTIONS K. MATOMÄKI Department of Mathematics, University of Turu,
More informationOn distribution functions of ξ(3/2) n mod 1
ACTA ARITHMETICA LXXXI. (997) On distribution functions of ξ(3/2) n mod by Oto Strauch (Bratislava). Preliminary remarks. The question about distribution of (3/2) n mod is most difficult. We present a
More informationOn some lower bounds of some symmetry integrals. Giovanni Coppola Università di Salerno
On some lower bounds of some symmetry integrals Giovanni Coppola Università di Salerno www.giovannicoppola.name 0 We give lower bounds of symmetry integrals I f (, h) def = sgn(n x)f(n) 2 dx n x h of arithmetic
More informationErgodic aspects of modern dynamics. Prime numbers in two bases
Ergodic aspects of modern dynamics in honour of Mariusz Lemańczyk on his 60th birthday Bedlewo, 13 June 2018 Prime numbers in two bases Christian MAUDUIT Institut de Mathématiques de Marseille UMR 7373
More informationOn prime factors of integers which are sums or shifted products. by C.L. Stewart (Waterloo)
On prime factors of integers which are sums or shifted products by C.L. Stewart (Waterloo) Abstract Let N be a positive integer and let A and B be subsets of {1,..., N}. In this article we discuss estimates
More informationPossible Group Structures of Elliptic Curves over Finite Fields
Possible Group Structures of Elliptic Curves over Finite Fields Igor Shparlinski (Sydney) Joint work with: Bill Banks (Columbia-Missouri) Francesco Pappalardi (Roma) Reza Rezaeian Farashahi (Sydney) 1
More informationOn sums of squares of primes
Under consideration for publication in ath. Proc. Camb. Phil. Soc. 1 On sums of squares of primes By GLYN HARAN Department of athematics, Royal Holloway University of London, Egham, Surrey TW 2 EX, U.K.
More informationDiophantine Approximation by Cubes of Primes and an Almost Prime
Diophantine Approximation by Cubes of Primes and an Almost Prime A. Kumchev Abstract Let λ 1,..., λ s be non-zero with λ 1/λ 2 irrational and let S be the set of values attained by the form λ 1x 3 1 +
More informationOn Some Mean Value Results for the Zeta-Function and a Divisor Problem
Filomat 3:8 (26), 235 2327 DOI.2298/FIL6835I Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat On Some Mean Value Results for the
More informationHOW OFTEN IS EULER S TOTIENT A PERFECT POWER? 1. Introduction
HOW OFTEN IS EULER S TOTIENT A PERFECT POWER? PAUL POLLACK Abstract. Fix an integer k 2. We investigate the number of n x for which ϕn) is a perfect kth power. If we assume plausible conjectures on the
More informationRemarks on the Pólya Vinogradov inequality
Remarks on the Pólya Vinogradov ineuality Carl Pomerance Dedicated to Mel Nathanson on his 65th birthday Abstract: We establish a numerically explicit version of the Pólya Vinogradov ineuality for the
More informationCongruences and exponential sums with the sum of aliquot divisors function
Congruences and exonential sums with the sum of aliquot divisors function Sanka Balasuriya Deartment of Comuting Macquarie University Sydney, SW 209, Australia sanka@ics.mq.edu.au William D. Banks Deartment
More informationNUMBER FIELDS WITHOUT SMALL GENERATORS
NUMBER FIELDS WITHOUT SMALL GENERATORS JEFFREY D. VAALER AND MARTIN WIDMER Abstract. Let D > be an integer, and let b = b(d) > be its smallest divisor. We show that there are infinitely many number fields
More informationFirst, let me recall the formula I want to prove. Again, ψ is the function. ψ(x) = n<x
8.785: Analytic Number heory, MI, spring 007 (K.S. Kedlaya) von Mangoldt s formula In this unit, we derive von Mangoldt s formula estimating ψ(x) x in terms of the critical zeroes of the Riemann zeta function.
More informationProofs of the infinitude of primes
Proofs of the infinitude of primes Tomohiro Yamada Abstract In this document, I would like to give several proofs that there exist infinitely many primes. 0 Introduction It is well known that the number
More informationNORMAL GROWTH OF LARGE GROUPS
NORMAL GROWTH OF LARGE GROUPS Thomas W. Müller and Jan-Christoph Puchta Abstract. For a finitely generated group Γ, denote with s n(γ) the number of normal subgroups of index n. A. Lubotzky proved that
More informationCONSECUTIVE PRIMES AND BEATTY SEQUENCES
CONSECUTIVE PRIMES AND BEATTY SEQUENCES WILLIAM D. BANKS AND VICTOR Z. GUO Abstract. Fix irrational numbers α, ˆα > 1 of finite type and real numbers β, ˆβ 0, and let B and ˆB be the Beatty sequences B.=
More informationON THE DIVISOR FUNCTION IN SHORT INTERVALS
ON THE DIVISOR FUNCTION IN SHORT INTERVALS Danilo Bazzanella Dipartimento di Matematica, Politecnico di Torino, Italy danilo.bazzanella@polito.it Autor s version Published in Arch. Math. (Basel) 97 (2011),
More informationA Subrecursive Refinement of FTA. of the Fundamental Theorem of Algebra
A Subrecursive Refinement of the Fundamental Theorem of Algebra P. Peshev D. Skordev Faculty of Mathematics and Computer Science University of Sofia Computability in Europe, 2006 Outline Introduction 1
More informationDABOUSSI S VERSION OF VINOGRADOV S BOUND FOR THE EXPONENTIAL SUM OVER PRIMES (slightly simplified) Notes by Tim Jameson
DABOUSSI S VERSION OF VINOGRADOV S BOUND FOR THE EXPONENTIAL SUM OVER PRIMES (slightly simplified) Notes by Tim Jameson (recalled on a busman s holiday in Oslo, 00) Vinogradov s method to estimate an eponential
More informationMath 259: Introduction to Analytic Number Theory Exponential sums II: the Kuzmin and Montgomery-Vaughan estimates
Math 59: Introduction to Analytic Number Theory Exponential sums II: the Kuzmin and Montgomery-Vaughan estimates [Blurb on algebraic vs. analytical bounds on exponential sums goes here] While proving that
More informationNEWMAN S INEQUALITY FOR INCREASING EXPONENTIAL SUMS
NEWMAN S INEQUALITY FOR INCREASING EXPONENTIAL SUMS Tamás Erdélyi Dedicated to the memory of George G Lorentz Abstract Let Λ n := {λ 0 < λ < < λ n } be a set of real numbers The collection of all linear
More informationProducts of ratios of consecutive integers
Products of ratios of consecutive integers Régis de la Bretèche, Carl Pomerance & Gérald Tenenbaum 27/1/23, 9h26 For Jean-Louis Nicolas, on his sixtieth birthday 1. Introduction Let {ε n } 1 n
More informationShort Kloosterman Sums for Polynomials over Finite Fields
Short Kloosterman Sums for Polynomials over Finite Fields William D Banks Department of Mathematics, University of Missouri Columbia, MO 65211 USA bbanks@mathmissouriedu Asma Harcharras Department of Mathematics,
More informationPATTERNS OF PRIMES IN ARITHMETIC PROGRESSIONS
PATTERNS OF PRIMES IN ARITHMETIC PROGRESSIONS JÁNOS PINTZ Rényi Institute of the Hungarian Academy of Sciences CIRM, Dec. 13, 2016 2 1. Patterns of primes Notation: p n the n th prime, P = {p i } i=1,
More informationThe zeros of the derivative of the Riemann zeta function near the critical line
arxiv:math/07076v [mathnt] 5 Jan 007 The zeros of the derivative of the Riemann zeta function near the critical line Haseo Ki Department of Mathematics, Yonsei University, Seoul 0 749, Korea haseoyonseiackr
More informationA 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 informationOn a combinatorial method for counting smooth numbers in sets of integers
On a combinatorial method for counting smooth numbers in sets of integers Ernie Croot February 2, 2007 Abstract In this paper we develop a method for determining the number of integers without large prime
More informationWilson s theorem for block designs Talks given at LSBU June August 2014 Tony Forbes
Wilson s theorem for block designs Talks given at LSBU June August 2014 Tony Forbes Steiner systems S(2, k, v) For k 3, a Steiner system S(2, k, v) is usually defined as a pair (V, B), where V is a set
More informationArithmetic progressions in sumsets
ACTA ARITHMETICA LX.2 (1991) Arithmetic progressions in sumsets by Imre Z. Ruzsa* (Budapest) 1. Introduction. Let A, B [1, N] be sets of integers, A = B = cn. Bourgain [2] proved that A + B always contains
More informationFermat 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 informationTHE SUM OF DIGITS OF n AND n 2
THE SUM OF DIGITS OF n AND n 2 KEVIN G. HARE, SHANTA LAISHRAM, AND THOMAS STOLL Abstract. Let s q (n) denote the sum of the digits in the q-ary expansion of an integer n. In 2005, Melfi examined the structure
More informationarxiv: v1 [math.nt] 5 Mar 2015
ON SUMS OF FOUR SQUARES OF PRIMES ANGEL KUMCHEV AND LILU ZHAO arxiv:1503.01799v1 [math.nt] 5 Mar 2015 Let Abstract. Let E(N) denote the number of positive integers n N, with n 4 (mod 24), which cannot
More informationFlat primes and thin primes
Flat primes and thin primes Kevin A. Broughan and Zhou Qizhi University of Waikato, Hamilton, New Zealand Version: 0th October 2008 E-mail: kab@waikato.ac.nz, qz49@waikato.ac.nz Flat primes and thin primes
More informationPrimary Decompositions of Powers of Ideals. Irena Swanson
Primary Decompositions of Powers of Ideals Irena Swanson 2 Department of Mathematics, University of Michigan Ann Arbor, Michigan 48109-1003, USA iswanson@math.lsa.umich.edu 1 Abstract Let R be a Noetherian
More informationON THE ITERATES OF SOME ARITHMETIC. P. Erd~s, Hungarian Academy of Sciences M. V. Subbarao, University of Alberta
ON THE ITERATES OF SOME ARITHMETIC FUNCTIONS. Erd~s, Hungarian Academy of Sciences M. V. Subbarao, University of Alberta I. Introduction. For any arithmetic function f(n), we denote its iterates as follows:
More informationNON-LINEAR COMPLEXITY OF THE NAOR REINGOLD PSEUDO-RANDOM FUNCTION
NON-LINEAR COMPLEXITY OF THE NAOR REINGOLD PSEUDO-RANDOM FUNCTION William D. Banks 1, Frances Griffin 2, Daniel Lieman 3, Igor E. Shparlinski 4 1 Department of Mathematics, University of Missouri Columbia,
More informationOn the Uniform Distribution of Certain Sequences
THE RAANUJAN JOURNAL, 7, 85 92, 2003 c 2003 Kluwer Academic Publishers. anufactured in The Netherlands. On the Uniform Distribution of Certain Sequences. RA URTY murty@mast.queensu.ca Department of athematics,
More informationThe Degree of the Splitting Field of a Random Polynomial over a Finite Field
The Degree of the Splitting Field of a Random Polynomial over a Finite Field John D. Dixon and Daniel Panario School of Mathematics and Statistics Carleton University, Ottawa, Canada {jdixon,daniel}@math.carleton.ca
More informationNumber Theoretic Designs for Directed Regular Graphs of Small Diameter
Number Theoretic Designs for Directed Regular Graphs of Small Diameter William D. Banks Department of Mathematics, University of Missouri Columbia, MO 65211 USA bbanks@math.missouri.edu Alessandro Conflitti
More informationOn Diophantine m-tuples and D(n)-sets
On Diophantine m-tuples and D(n)-sets Nikola Adžaga, Andrej Dujella, Dijana Kreso and Petra Tadić Abstract For a nonzero integer n, a set of distinct nonzero integers {a 1, a 2,..., a m } such that a i
More informationSplitting Fields for Characteristic Polynomials of Matrices with Entries in a Finite Field.
Splitting Fields for Characteristic Polynomials of Matrices with Entries in a Finite Field. Eric Schmutz Mathematics Department, Drexel University,Philadelphia, Pennsylvania, 19104. Abstract Let M n be
More informationEVERY NATURAL NUMBER IS THE SUM OF FORTY-NINE PALINDROMES
#A3 INTEGERS 16 (2016) EVERY NATURAL NUMBER IS THE SUM OF FORTY-NINE PALINDROMES William D. Banks Department of Mathematics, University of Missouri, Columbia, Missouri bankswd@missouri.edu Received: 9/4/15,
More informationEkkehard Krätzel. (t 1,t 2 )+f 2 t 2. Comment.Math.Univ.Carolinae 43,4 (2002)
Comment.Math.Univ.Carolinae 4,4 755 77 755 Lattice points in some special three-dimensional convex bodies with points of Gaussian curvature zero at the boundary Eehard Krätzel Abstract. We investigate
More informationThe least prime congruent to one modulo n
The least prime congruent to one modulo n R. Thangadurai and A. Vatwani September 10, 2010 Abstract It is known that there are infinitely many primes 1 (mod n) for any integer n > 1. In this paper, we
More informationON SUMS OF PRIMES FROM BEATTY SEQUENCES. Angel V. Kumchev 1 Department of Mathematics, Towson University, Towson, MD , U.S.A.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A08 ON SUMS OF PRIMES FROM BEATTY SEQUENCES Angel V. Kumchev 1 Department of Mathematics, Towson University, Towson, MD 21252-0001,
More informationOleg Eterevsky St. Petersburg State University, Bibliotechnaya Sq. 2, St. Petersburg, , Russia
ON THE NUMBER OF PRIME DIVISORS OF HIGHER-ORDER CARMICHAEL NUMBERS Oleg Eterevsky St. Petersburg State University, Bibliotechnaya Sq. 2, St. Petersburg, 198904, Russia Maxim Vsemirnov Sidney Sussex College,
More informationSIX PROBLEMS IN ALGEBRAIC DYNAMICS (UPDATED DECEMBER 2006)
SIX PROBLEMS IN ALGEBRAIC DYNAMICS (UPDATED DECEMBER 2006) THOMAS WARD The notation and terminology used in these problems may be found in the lecture notes [22], and background for all of algebraic dynamics
More informationON 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 informationMultiplicative Order of Gauss Periods
Multiplicative Order of Gauss Periods Omran Ahmadi Department of Electrical and Computer Engineering University of Toronto Toronto, Ontario, M5S 3G4, Canada oahmadid@comm.utoronto.ca Igor E. Shparlinski
More informationRank-one Twists of a Certain Elliptic Curve
Rank-one Twists of a Certain Elliptic Curve V. Vatsal University of Toronto 100 St. George Street Toronto M5S 1A1, Canada vatsal@math.toronto.edu June 18, 1999 Abstract The purpose of this note is to give
More informationSquares in products with terms in an arithmetic progression
ACTA ARITHMETICA LXXXVI. (998) Squares in products with terms in an arithmetic progression by N. Saradha (Mumbai). Introduction. Let d, k 2, l 2, n, y be integers with gcd(n, d) =. Erdős [4] and Rigge
More informationarxiv: v1 [math.co] 22 May 2014
Using recurrence relations to count certain elements in symmetric groups arxiv:1405.5620v1 [math.co] 22 May 2014 S.P. GLASBY Abstract. We use the fact that certain cosets of the stabilizer of points are
More informationOn Roth's theorem concerning a cube and three cubes of primes. Citation Quarterly Journal Of Mathematics, 2004, v. 55 n. 3, p.
Title On Roth's theorem concerning a cube and three cubes of primes Authors) Ren, X; Tsang, KM Citation Quarterly Journal Of Mathematics, 004, v. 55 n. 3, p. 357-374 Issued Date 004 URL http://hdl.handle.net/107/75437
More informationINDECOMPOSABLE GORENSTEIN MODULES OF ODD RANK. Christel Rotthaus, Dana Weston and Roger Wiegand. July 17, (the direct sum of r copies of ω R)
INDECOMPOSABLE GORENSTEIN MODULES OF ODD RANK Christel Rotthaus, Dana Weston and Roger Wiegand July 17, 1998 Abstract. Let (R,m) be a local Cohen-Macaulay ring with m-adic completion R. A Gorenstein R-module
More information#A2 INTEGERS 12A (2012): John Selfridge Memorial Issue ON A CONJECTURE REGARDING BALANCING WITH POWERS OF FIBONACCI NUMBERS
#A2 INTEGERS 2A (202): John Selfridge Memorial Issue ON A CONJECTURE REGARDING BALANCING WITH POWERS OF FIBONACCI NUMBERS Saúl Díaz Alvarado Facultad de Ciencias, Universidad Autónoma del Estado de México,
More informationOn the Diophantine equation k
On the Diophantine equation k j=1 jfp j = Fq n arxiv:1705.06066v1 [math.nt] 17 May 017 Gökhan Soydan 1, László Németh, László Szalay 3 Abstract Let F n denote the n th term of the Fibonacci sequence. Inthis
More informationSOME REMARKS ON NUMBER THEORY BY P. ERDŐS 1. Let ABSTRACT This note contains some disconnected minor remarks on number theory. (1) Iz j I=1, 1<j<co be
SOME REMARKS ON NUMBER THEORY BY P. ERDŐS 1. Let ABSTRACT This note contains some disconnected minor remarks on number theory. (1) Iz j I=1, 1
More informationINTEGRAL ORTHOGONAL BASES OF SMALL HEIGHT FOR REAL POLYNOMIAL SPACES
INTEGRAL ORTHOGONAL BASES OF SMALL HEIGHT FOR REAL POLYNOMIAL SPACES LENNY FUKSHANSKY Abstract. Let P N (R be the space of all real polynomials in N variables with the usual inner product, on it, given
More informationOn pseudosquares and pseudopowers
On pseudosquares and pseudopowers Carl Pomerance Department of Mathematics Dartmouth College Hanover, NH 03755-3551, USA carl.pomerance@dartmouth.edu Igor E. Shparlinski Department of Computing Macquarie
More informationA NEW UPPER BOUND FOR FINITE ADDITIVE BASES
A NEW UPPER BOUND FOR FINITE ADDITIVE BASES C SİNAN GÜNTÜRK AND MELVYN B NATHANSON Abstract Let n, k denote the largest integer n for which there exists a set A of k nonnegative integers such that the
More informationEquidivisible 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 informationResearch Statement. Enrique Treviño. M<n N+M
Research Statement Enrique Treviño My research interests lie in elementary analytic number theory. Most of my work concerns finding explicit estimates for character sums. While these estimates are interesting
More informationFibonacci numbers of the form p a ± p b
Fibonacci numbers of the form p a ± p b Florian Luca 1 and Pantelimon Stănică 2 1 IMATE, UNAM, Ap. Postal 61-3 (Xangari), CP. 8 089 Morelia, Michoacán, Mexico; e-mail: fluca@matmor.unam.mx 2 Auburn University
More informationLEVEL MATRICES G. SEELINGER, P. SISSOKHO, L. SPENCE, AND C. VANDEN EYNDEN
LEVEL MATRICES G. SEELINGER, P. SISSOKHO, L. SPENCE, AND C. VANDEN EYNDEN Abstract. Let n > 1 and k > 0 be fixed integers. A matrix is said to be level if all its column sums are equal. A level matrix
More informationCullen Numbers in Binary Recurrent Sequences
Cullen Numbers in Binary Recurrent Sequences Florian Luca 1 and Pantelimon Stănică 2 1 IMATE-UNAM, Ap. Postal 61-3 (Xangari), CP 58 089 Morelia, Michoacán, Mexico; e-mail: fluca@matmor.unam.mx 2 Auburn
More informationNew Examples of Noncommutative Λ(p) Sets
New Examples of Noncommutative Λ(p) Sets William D. Banks Department of Mathematics, University of Missouri Columbia, MO 6511 USA bbanks@math.missouri.edu Asma Harcharras Department of Mathematics, University
More informationOn the Fractional Parts of a n /n
On the Fractional Parts of a n /n Javier Cilleruelo Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM and Universidad Autónoma de Madrid 28049-Madrid, Spain franciscojavier.cilleruelo@uam.es Angel Kumchev
More information