A PROPERTY OF CERTAIN SQUARES

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1 I! A PROPERTY OF CERTAIN SQUARES TR95-00 JOHN H. HALTON University of North Carolina Chapel Hill, NC Department of Computer Science CB #375, Sitterson Hall UNC-Chapel Hill Chapel Hill, NC UNC is an Equal Opportunity/Affirmative Action Institution.

2 .. A PROPERTY. OF CERTAIN SQUARES JOHN H. HALTON University of North Carolina Chapel Hill, NC This note arises from the serendipitous observation that both = and = 38 are perfect squares. A further examination of the first few perfect squares and the numbers obtained by appending a digit to them yields the following: TABLE I =, 6 =, 0 = 00, =, 9 = 7, =, 3 = 9, =, = 6, 69 = 3, 3 = 69, 5 = 5, 56 = 6, = 96, 6 = 36, 36 = 9, 5 = 5 7 = 9, 6 = 56 8 = 6, 7 = 89 9 = 8, 8 = 3 39 = 57 The question now arises, whether there are any more such pairs of perfect squares, and what regularities apply. Consider the general case. The Diophantine equation (with integer x, y, and b) is 0x + b = y, 0 ~ b ~ 9, x ~ 0, y ~ 0. () (i) If x = 0, the only solutions are x = 0, with (b, y) = (0, 0), (, ), (, ), or (9, 3). (ii) If x =, the only solution clearly is X =, y =, b = 6. (iii) If x ~, since 0 is not a perfect square, b :t:. 0. Furthermore, let us write (with integer u and v) () (3) then or y = lou + v, - ~ v ~ 5, 0x + b = 00u + 0uv + v, 0(x -0u - uv) = v- b. () (5)

3 J. H. HALTON A PROPERTYOFCERTAIN5QUARES Since the left-hand side is divisible by 0, if b = 5, then necessarily v = 5, and then x = 0u(u- ) +, which is impossible; because the last digit of a perfect square cannot be. Thus, b * 5. For v- b to be divisible by 0, the only remaining possibilities are Let us put that is (with integer z and real~, (b, v) = (, ±), (, ±), (6, ±), (9, ±3). z = lxot (6) (7a) so that and (with integer c) so that and so Then, by (), (7b), and (8a), whence, by (6), (7c), and (8b), x{lo = z +,, 0 < ' <, 3x ~ z < x; y = z + c, c >,, c ~. (z + ~ + b = (z + c), (7b) (7c) (8a) (8b) (8c) (9) 6x(c- ~ ~ z(c - ~ = b - C + ~ < b ~ 9. If x ~ and c ~, then c-' > c- ~, and so (0) becomes < 9, a contradiction. Thus, with (8c), and by (3), (0) if x ~, then c =. () TABLE II X b z c / ' --

4 J.H.HALTON A PROPERTY OF CERTAIN SQUARES Table II above shows the parameters for the solutions given in Table I. If we put ' = - ], () and, by (Sa) with (), z = y-, (3) then, by (7b), r{io = y- ], 0 < ] <. i.e., This establishes the following result. y = lx{iqi. () (5) Theorem. For every integer x ~, the fraction ~ is the rational least upper bound of -{0 having x as denominator. It follows from (5) that there is a simple algorithm for finding possible solutions of (). All that is necessary is, for each possible integer value of x, to compute y by applying (5), and b from (), in the form b = y - 0x > 0. (6) If the resulting value of b is less than 0, we have an acceptable solution to our problem. For example, consider the three consecutive cases: TABLE ill X y x y b Only x = 80 yields an acceptable solution. Using this algorithm, all the acceptable solutions of the Diophantine equation () for 0 ~ x ~ 500,000 have been computed and are tabulated below. -3-

5 J.H.HALTON A PROPERTY OF CERTAIN SQUARES TABLE IV n X y x y b We immediately observe that there appears to be a periodicity in the values of the incremental digit b: the period takes the form ",, 9, 6, 9, 9, 6/' in the data we have. This feature has been emphasized in the table by drawing a line above the data with b =. In each period, the values and occur only once, while 6 occurs twice and 9 occurs thrice. - -

6 ].H. HALTON A PROPERlYOFCERTAINSQUARES At this point, bearing in mind the result embodied in Theorem, it seemed appropriate to examine the literature of this problem. The search began with the discussion of continued fractions in Hardy & Wright (HAR6), thence moved on to Chrystal (CHR6), and so to Dickson (DIC5) and Gelfond (GEL6). It is found in the literature that the equation y- 0x = (7) will have infinitely many solutions; and that, if the solution (x, y ) is minimal, in the sense that y + x -{i0 = min { y +.rjio : x > 0, y > 0, y -0x = }, (8) then all the solutions of (7) take the form ~-pn xn = -{0 ' Yn = (9) where a = Y + xl-{0, f3 = Y- xl{io. (0) We can readily derive from this that, in our case, x = 6, y = 9, and so Now consider a recurrence relation of the form a = {0 z , (a) f3 = 9-6-{0 z (b) This will apply to both the xn and the Yn given in (9), if we can find coefficients A and B, such that, for all n, both a and f3 satisfy A_.n+ = AA.n+ + BA.n, i.e., (3) Substituting (), we see that this is indeed possible, when 7 ± 8-{0 = A(9 ± 6-{0) + B, () See Chrystal (CHR6), Chaps. 3 and 33; Dickson-where a further, very extensive, historical bibliography is supplied-(dic5), Chap. ; Gelfond (GEL6), Chaps. and 5; and Hardy & Wright (HAR6), Chap. 0. A very careful analysis of y'- axz =, a generalized form of (7), is given by Gelfond. This equation is often referred to as Pell's equation; but appears to have been first considered by Fennat. -5-

7 J.H.HALTON A PROPERTY OF CERTAIN SQUARES and these two equations have the unique solution A = 38, B = -. Thus, both the xn and the Yn satisfy the recurrence () sn+ = 38sn+l- sn. We can verify that these relations are borne out by the solutions tabulated on page above, for the b = entries only. Thus, we have the following result. Theorem. (5) All the integer solutions of () when b = are given by the recurrence (5), with initial values x 0 = 0, x = 6, Yo =, y = 9. It is interesting to note that all the x andy entries in Table IV appear to satisfy the relation (5), when one skips from each entry to the seventh entry down the table as successor. Now suppose that (x, y) is any Diophantine (integer) solution of () with b = and that (p, q) is any Diophantine solution of () with b :t= (that is, by (6), with b chosen arbitrarily to be, 6, or 9). Let us write u = qx + py, } v = 0px + qy. (6) Then v - 0u = (0px + qy)- 0(qx + py) = (00p - 0q)x + (q- 0p)y = (q- 0p)(y- 0x) = b x = b; (7) so that (u, v) is a solution of () with the given value of b; and, clearly, u and v will be integers, since p, q, x, andy are integers. Furthermore, if we consider (x, y) = (xw Yn), (xn+l' Yn+l), and (xn+' Yn+), related by (5), then, by (6), the corresponding pairs (u, v) = (uw vn), (un+l' vn+l), and (un+, vn+), will be related by (8) (9) un+ = qxn+ + PYn+ = q(38xn+l - xn) + p(38yn+l - Yn) and = 38(qxn+l + PYn+l)- (qxn + PYn) = 38un+l- un' (30a) vn+ = 0pxn+ + qyn+ = 0p(38xn+l - xn) + q(38yn+l-yn) = 38(0pxn+l + qyn+l)- (0pxn + qyn) = 38vn+l- vn. (30b) -6-

8 J.H.HALTON A PROPERTYOFCERTA.W5QUARES That is to say, starting with any integer solution (p, q) of () with given integer b, we can generate an infinite sequence of such solutions (un, vn), in one-to-one correspondence with the solutions (x"' Yn) of () with b =. All these sequences satisfy the same recurrence, (5). Note that this is equivalent to so that the recurrence proceeds without end, both forward and backward (i.e., both as n increases and as it decreases), yielding integer solutions. Also, taking (x, y) = (x 0, y 0 ) = (0, ), we see that Substituting (9) in (6), we get that (3) (u 0, v 0 ) = (p, q). (3) q + p-{wj n (q -p-{wj un = qxn + PYn = ( -{0 a - -{0 pn (33a) and ( vn = 0pxn + qyn = Thus we get the following result. q + p-{wj n (q - p-{wj a + /Y', (33b) Theorem 3. Given any integer solution (p, q) of ( ) with given b, we can generate an infinite sequence of such integer solutions (un, vn), given explicitly by (33). These solutions are in one-to-one correspondence, given by (6), with the solutions (x"' Yn) of () with b =. It is easily verified from (9) that Hence, by (33), i.e., and Xm+n = XrrJ!n + XnYm' Ym+n = Yrr8n + 0xmxn. um+n = q(xmyn + XnYm) + p(ymyn + 0xmxn), (3) (35a) i.e., (35b) From this we obtain the following result. Theorem. We can replace (p, q) by any member of the sequence of pairs (urn, vm) in (6) and (33) without changing the sequence of solutions defined in Theorem

9 }.H. HALTON A PROPERTY OF C ERT AlN SQUARES Now, since, by (), q - 0p = b > 0, and we assume p ~ 0 and q ~ 0, it follows that q +pfw ~ q-p{w > 0. (36) We can write (33) as un = J.Ui - vfj", vn = {0 (Jld + vf!'), (37) with q+pfw q-p{w :!:. J = {0 ~ v = {0 > 0, v ~. (38) (Equality occurs in (38), if and only if p = 0.) Therefore, by (), and since there will be a unique integer m ~ 0, such that af3 =, (39) ' and then _.?m. am+ > ::!:. v > - cr, (0) i.e., (a) while i.e., J.Lcrm- < vp-m-, u- m- < 0. Furthermore, by (), (37), and (38), un+ - un = J.Lct" (a -) + v/3"( - /3) > 0, (b) () so that the sequence of un is strictly monotone-increasing, and so un ~ 0 if and only if n ~ - m. (3) We have thus shown that there will be a solution (u_m, v_m) with minimal positive first component (the second component is always positive). Also, by () and (38), () Thus, by (0),,m >... ~ ~ - -'J Ji (5a) I and cr m- < ~ Jl' so that a-m- < -'J... ~whence Ji I -8-

10 .' '. J. H. HALTON A PROPERTY OF CERTAIN SQUARES crm <a~ (5b) Thus, by (37), (5b), (39), (), and (), u_m = JlCXm- vjrm < a~- [3~ = (a- [3) ~ i.e., ;...JiQ X "!fw ; 6-./b ~ 6; (6) ' This yields the following theorem. Theorem 5. a member between x 0 = 0 and x = 6. The sequence of solutions specified in Theorem 3 contains It follows that we can always adopt (p, q) as the member (u_m' v_m) defined above, which lies in (0, 6). Let us do this, from now on; so that the member of the sequence (un' vn) with minimal non-negative first component will be identified as (u 0, v 0 ) = (p, q). This proves the following theorem. Theorem 6. All the integer solutions of () with y > 0 are identified as the sequences defined in Theorem 3, with (p, q) = (0, ) for b =, (0, ) for b =, (0, 3) or (, 7) or (, 3) for b = 9, (, ) or (5, 6) for b = 6 Proof. Theorem restates the known result, that all integer solutions of () when b = are given by (9) with (0), or, alternatively, by the sequence satisfying (5), with initial values (0, ) and (6, 9). Theorem 3 tells us that, if (p, q) is any integer solution of () with some other allowable value of b (i.e., by (6), one of, 6, and 9), then we can generate a sequence of solutions (un' vn) which correspond one-to-one to the (xn' Yn), via (6) or (33). Theorem tells us that there will always be a member of such a sequence lying strictly between x 0 = 0 and x = 6. Therefore, by inspection of Table IV, we see that the only such sequences possible are those enumerated in (7). 3 (7) 3 There is the additional, isolated case of x = 0, y = 0, b = O-see ()-but this leads to no other solutions; because the general solution (37) has J.l = v =

11 }.H. HALTON A PROPERTY OF CERTAJN SQUARES Now, let us suppose that, for given n, (xn' Yn), (un, vn), and (r n' tn) are all integer solutions of (), belonging to sequences defined as in (9), (0), and (33), and that (8a) and Then, by (33), 0 = x 0 ~ u 0 ~ r 0 < x = 6 = Yo < vo < to < Y = 9. (8b) or, by (3), (9a) or (9b) This last result yields the following. Theorem 7. The seven sequences of integer solutions of () for various b, developed in Theorems -6 are interlaced, in the sense that each sequence retains its member-by-member order, relative to the others. In other words, if we list solutions (as in Table IV) in order of increasing y-values, every seventh entry belongs to the same sequence. We have.thus proved that the conjectures generated by examination of Table IV are universally true for the Diophantine problem (). Now let us generalize this to ax + b = y, 0 ~ b ~ a-, x ~ 0, y ~ 0. (50) As was mentioned in Footnote, so long as a is not a perfect square, the equation with b =, is known to have infinitely many integer solution (x, y), all of them taking the form 5 ci"-/y' ci"+/y' (5) xn = -{i ' Yn = (5) where 6 (53) 5 6 Compare (7). Compare (9). Compare (0). t - 0 -

12 J.H.HALTON A PROPERTY OF CERTAIN SQUARES and (x, y ) is the (unique) solution which minimizes y + x{;. The entire argument presented above for a = 0 clearly generalizes to any a which is not a perfect square? Thus, Theorems -7 hold for any such a, not just for a = 0, mutatis mutandis, as to the particular solution (x, y ). For instance, if a = 7, then 0 ~ b ~ 6. If x = 0, then the only acceptable solutions have (b, y) = (0, 0), (, ), and (, ); if x =, then (b, y) = (, 3) is the only solution; and if x ~, then, putting 8 y = 7u + v, - 3 ~ v 5 3, (5) ' we get 9 7(x- 7u - uv) = v- b. Since the left-hand side is divisible by 7, it is easy to verify by enumeration that the only possibilities are (b, v) = (, ±), (, ±3), and (, ±). The argument in (7)-(5), now withlo (56a) r[7 = z +,, 0 < ' <, so that x ~ z < 3x; and (with integer c) y = z + c, (55) (56b) (8a) yields x(c- 0 ~ z(c- 0 = b- c + ( < b ~, which recovers that, if x ~, then c =. Hence, y = lrfll, yielding the appropriate version of Theorem. Table IV is replaced by Table V, shown below. We again observe a periodicity in the values of b, of the form ",,." This feature has been emphasized in the table by drawing a line above the data with b =. (57) () (58) The irrationality of ;/a is used in proving the result embodieed in (5)-(53). Compare (). Compare (5). Compare (7b), (7c). Compare (5). - -

13 J.H. HALTON A PROPERTY OF CERTAIN SQUARES TABLEV n X y x This shows that,' for a = 7, we have y b xl = 3, Yt = 8, and so, by (53), a = {7 = and f3 = 8-3-{7;::: (59) (60a) (60b) Theorems - 7 now follow, just as before, with appropriate modifi~ations. --

14 , J. H. HALTON A PROPERTY OF CERTAJN SQUARES.. CHR6 DIC5 GEL6 HAR6 REFERENCES G. CHRYSTAL, Algebra-An Elementary Text-book for the Higher Classes of Secondary Schools and for Colleges. Volume, Seventh Edition, Chelsea Publishing Co., New York, 96, 68 pp. L. E. DICKSON, History of the Theory of Numbers. Volume, Diophantine Analysis. Seventh Edition, Chelsea Publishing Co., New York, 95, 803 pp. A. 0. G ELFOND, The Solution of Equations in Integers. W. H. Freeman & Co., San Francisco, 96, 63 pp. G. H. HARDY, E. M. WRIGHT, An Introduction to the Theory of Numbers. Fourth Edition, Corrected, Clarendon Press, Oxford, 96, pp. -, - 3-

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