On systems of Diophantine equations with a large number of integer solutions

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1 On systems of Diophantine equations with a large number of integer solutions Apoloniusz Tysza Abstract arxiv: v [math.nt] 26 Oct 205 Let E n = {x i + x j = x, x i x j = x : i, j, {,..., n}}. For each integer n 3, J. Browin defined a system B n E n which has exactly b n solutions in integers x,..., x n, where b n N \ {0} and the sequence {b n } n=3 rapidly tends to infinity. For each integer n 2, we define a system T n E n which has exactly t n solutions in integers x,..., x n, t n where t n N \ {0} and lim =. n b n 200 Mathematics Subject Classification: Primary: D45; Secondary: D72, E25. Key words and phrases: large number of integer solutions, number of representations of n as a sum of three squares of integers, system of Diophantine equations. For a non-negative integer n, let r n) denote the number of representations of n as a sum of squares of integers. Let E n = {x i + x j = x, x i x j = x : i, j, {,..., n}}. For an integer n 3, let B n denote the following system of equations []): i {,..., n 3} x i x i = x i+ x n + x n = x x n x n = x x n 0 x n 0 = x n 9 x n 8 x n 8 = x n 7 x n 6 x n 6 = x n 5 x n 4 x n 4 = x n 3 x n 9 + x n 7 = x n 2 x n 5 + x n 3 = x n x n 2 + x n = x n x n + x n 2 = x n The system B n is contained in E n and B n equivalently expresses that x =... = x n = 0 or x n = 2) ) x n = n 2 = x 2 n 0 + x2 n 8 + x2 n 6 + x2 n 4 all the other variables are uniquely determined by the above con junction and the equations o f B n.

2 Theorem. []) The system B n has exactly + r n 2) = + 8σ n 2) solutions in integers x,..., x n, where σ ) denote the sum of positive divisors. For a positive integer n 2, let T n denote the following system of equations: i {,..., n 2} x i x i = x i+ x n 0 x n 0 = x n 0 x n 0 + x n 0 = x n 9 x n 8 + x n 9 = x x n 8 x n 7 = x n x n 0 x n 7 = x n 7 x n 6 x n 6 = x n 5 x n 4 x n 4 = x n 3 x n 2 x n 2 = x n x n 5 + x n 3 = x n x n + x n = x n Theorem 2. The system T n is contained in E n and T n has exactly + 2n 2 2 r ) 2 n 2) 3 ± 2 = 0 solutions in integers x,..., x n. Proof. The system T n equivalently expresses that or x n 0 = ) x =... = x n = 0 ) x 2) x n 7 = x n = x 2n 2 = x 2 n 6 + x2 n 4 + x2 n 2 all the other variables are uniquely determined by the above con junction and the equations o f T n. In the second case, by the polynomial identity we obtain that x 2n 2 = 2 2n 2 + x 2) 2 2n 2 = x 2) x n 7 2 n 2 = 0 2 n 2 = 0 2 2n 2 x 2 2n 2 x Hence, x 2 divides 2 2n 2. Therefore, x { 2 ± 2 : [ 0, 2 n 2] Z }. Consequently, { 2 x n = x 2n 2 ) ± 2 2 n 2 : [ 0, 2 n 2] } Z Since the last five equations of T n equivalently express that x n = x 2 n 6 + x2 n 4 + x2 n 2, the proof is complete. The following lemma is a consequence of Siegel s theorem [4]). 2

3 Lemma. [2, p. 9], [3, p. 27]) For every ε 0, ) there exists c ε) 0, ) such that ) r 3 4 s m c ε) m ε 2 for every non-negative integer s and every positive integer m {4 : Z} {8 + 7: Z}. Let b n denote the number of integer solutions of B n, and let t n denote the number of integer solutions of T n. t n Theorem 3. lim =. n b n Proof. Let an integer n is greater than 2. By Theorem 2, t n > r n 2) 2 n 2 = r 3 42n n 2 ) 2 n 2 By Lemma, for each ε 0, ) there exists cε) 0, ) such that r 3 42n n 2 ) 2 n 2 cε) + 2 2n 2 ) 2 n 2 ε 2 for every integer n > 2. We tae ε = 4. Since 2n 2 2 n 3, we get cε) + 2 2n 2 ) 2 n 2 2 ε > c ) 2 2n 3) 2 n 2 ) 4 = c 2 22n Since < σ n 2) and n 2 < 2 2n, Theorem gives: b n = + 8σ n 2) < 9σ n 2) n 2 9 = 2 2n < 9 = < 9 2 2n 2 2n = 9 2 2n 0 < 2 2n 9 ) Therefore, t c 2 22n 27 n 4 >. This quotient tends to infinity when n tends to infinity, which b n 2 2n 9 completes the proof. 3

4 The following Mathematica code first computes decimal approximations of t n b n for all integers n [3, 9]. The output results show that t n > b n for every integer n [4, 9]. By Theorem 2, t 20 > r ) = r ) 256 ) The last command of the code finds the decimal approximation of the last quotient. Hence, t 20 > It seems that t n > b n for every integer n 2, although this remains unproven. Let us define the height of a rational number p by max p, q ) provided p is written in q q lowest terms. Let us define the height of a rational tuple x,..., x n ) as the maximum of n and the heights of the numbers x,..., x n. For an integer n 4, let S n denote the following system of equations: i {,..., n 4} x i x i = x i+ x n 2 + = x x n + = x n 2 x n x n = x n 3 Theorem 4. [5]) The system S n is soluble in positive integers and has only finitely many integer solutions. Each integer solution x,..., x n ) satisfies x,..., x n n 4) 2 n 4. The following equalities i {,..., n 3} x i = n 4) 2 i x n 2 = + 2 2n 4 x n = 2 2n 4 x n = + 2 2n 4 ) 2 n 4 define the unique integer solution whose height is maximal. 4

5 Conjecture. If an integer n is sufficiently large and a system U {x i + = x, x i x j = x : i, j, {,..., n}} has only finitely many solutions in positive integers x,..., x n, then each such solution x,..., x n ) satisfies x,..., x n n 4) 2 n 4. A bit stronger version of the Conjecture appeared in [5]. The Conjecture implies that there is an algorithm which taes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer non-negative integer, positive integer, rational) solutions, if the solution set is finite [5]). Let us pose the following two questions: Question. Is there an algorithm which taes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer solutions, if the solution set is finite? Question 2. Is there an algorithm which taes as input a Diophantine equation, returns an integer, and this integer is greater than the number of integer solutions, if the solution set is finite? Obviously, a positive answer to Question implies a positive answer to Question 2. Lemma 2. Let d denote the maximal height of an integer solution of a Diophantine equation Dx,..., x n ) = 0 whose solution set in integers is non-empty and finite. We claim that the number of integer solutions to the equation D 2 x,..., x n ) + n 2 + x x2 n u 2 u2 2 u2 3 u2 4 v2 v2 2 v2 3 v2 4) 2 = 0 is finite and greater than d. Proof. There exists an integer tuple a,..., a n ) such that Da,..., a n ) = 0 and max n, a,..., a n ) = d. The equation n 2 + a a2 n = x + y has n 2 + a a2 n + solutions in non-negative integers x and y. Since n 2 + a a2 n + d 2 + > d, the claim follows from Lagrange s four-square theorem. Theorem 5. A positive answer to Question 2 implies a positive answer to Question. Proof. In order to compute an upper bound on the heights of integer solutions to a Diophantine equation Dx,..., x n ) = 0 with a finite number of integer solutions, we compute an upper bound on the number of integer solutions to the equation D 2 x,..., x n ) + n 2 + x x2 n u 2 u2 2 u2 3 u2 4 v2 v2 2 v2 3 v2 4) 2 = 0 By Lemma 2, this number is greater than the heights of integer solutions to Dx,..., x n ) = 0. 5

6 References [] J. Browin, On systems of Diophantine equations with a large number of solutions, Colloq. Math ), no. 2, [2] W. Freeden, Metaharmonic lattice point theory, Chapman and Hall/CRC, 20. [3] P. Michel, Analytic number theory and families of automorphic L-functions, in: Automorphic Forms and Applications eds. P. Sarna and F. Shahidi), IAS/Par City Math. Series, vol. 2, Amer. Math. Soc., Providence, RI, 2007, [4] C. L. Siegel, Über die Classenzahl quadratischer Zahlörper, Acta Arith. 936), no., 83 86; Gesammelte Abhandlungen, Bd. I, , Springer, Berlin-Heidelberg- New Yor, 966. [5] A. Tysza, A hypothetical way to compute an upper bound for the heights of solutions of a Diophantine equation with a finite number of solutions, Proceedings of the 205 Federated Conference on Computer Science and Information Systems eds. M. Ganzha, L. Maciasze, M. Paprzyci), Annals of Computer Science and Information Systems, vol. 5, , IEEE Computer Society Press, 205, Apoloniusz Tysza University of Agriculture Faculty of Production and Power Engineering Balica 6B, Kraów, Poland address: rttysza@cyf-r.edu.pl 6