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Brindza 1991 diophantine problems involving powers factorials
theorem_1: States that for every positive integer r there is n_0(r) such that no sum n_1! + ... + n_r! with n_0(r) < n_1 < ... < n_r is powerful; the paper gives no explicit value of n_0(r).
theorem_2: States that there is an effectively computable absolute constant C such that every solution of (p-1)! + a^(p-1) = p^k in positive integers a, k, p, with p > 2 prime, satisfies max{p, a, k} < C.
theorem_3: States that for a nonzero integer D every solution of x^2 + D = p^k in positive integers x, p, k with k, p > 1 satisfies k/log k < C_3(p log p + log|D|) p log p, with C_3 an effectively computable absolute constant.
B. Brindza, P. Erdos, On some diophantine problems involving powers and factorials. Journal of the Australian Mathematical Society (Series A) 51 (1991), 1-7. doi:10.1017/S1446788700033255. The publisher's PDF prints "© 1991 Australian Mathematical Society 0263-6115/91 $A2.00 + 0.00" at the foot of its first page and, on every page, the footer "https://doi.org/10.1017/S1446788700033255 Published online by Cambridge University Press", every other right reserved.
Motivated by Mahler's last question on squares of the form sum of e_i k^i with e_i in {0, 1}, the paper asks whether sum e_i i! = x^z with e_i in {0, 1}, finitely many e_i nonzero and z > 1 has only finitely many solutions, and observes that in this generality the question is hopeless. Theorem 1 proves the fixed-summand result: for every positive integer r there is n_0(r) such that no integer of the form n_1! + ... + n_r! with n_0 < n_1 < ... < n_r is powerful, that is, each such integer has a prime dividing it to the first power only; the proof combines an elementary product-of-primes inequality with a strong theorem on primes in short intervals that has no effective proof, so no explicit n_0(r) is given. Theorem 2 shows that all solutions of equation (6), (p-1)! + a^{p-1} = p^k in positive integers a, k, p with p > 2 prime, satisfy max{p, a, k} < C for an effectively computable absolute constant C; the proof uses Baker's method for the lower and upper bounds on k in (7), the upper one C_2 p^3 and the lower one much larger in p. Theorem 3 bounds the exponent in the Ramanujan-Nagell type equation x^2 + D = p^k: for nonzero integer D, every solution in positive integers x, p, k with k, p > 1 has k/log k < C_3 (p log p + log|D|) p log p for an effectively computable absolute constant C_3. Theorem 2 thus shows that (6) has only finitely many solutions, which is what the Erdős-Graham question quoted on p. 3 asks. Problem 1108 asks whether the sums of finitely many distinct factorials include only finitely many kth powers (k >= 2) and only finitely many powerful numbers; Theorem 1 covers only sums n_1! + ... + n_r! of a fixed number r of factorials with n_0(r) < n_1 < ... < n_r, so it settles neither question.
Source: https://doi.org/10.1017/S1446788700033255.
Bears on. #405: Theorem 2 (p. 4) bounds p, a and k in (p-1)! + a^(p-1) = p^k by one effective absolute constant, so the equation has finitely many solutions in all and hence for each odd prime p; it does not list them (theorem_2, with the upper bound on k from theorem_3), #1108: Theorem 1 (p. 2) shows that for each fixed r no sum n_1! + ... + n_r! with n_0(r) < n_1 < ... < n_r is powerful, hence none is a kth power; it says nothing about sums with a summand n! for n <= n_0(r), nor about the sums of every length at once, so it settles neither question (theorem_1).
Results. Page numbers are those printed in the journal, pp. 1--7.
- Theorem 1 (p. 2): for every positive integer r there is n_0(r) such that no sum n_1! + ... + n_r! with n_0(r) < n_1 < ... < n_r is powerful; no explicit value of n_0(r) is given.
- Theorem 2 (p. 4): all solutions of (p-1)! + a^(p-1) = p^k in positive integers a, k, p with p > 2 prime satisfy max{p, a, k} < C for an effectively computable absolute constant C, proved by Baker's method.
- Theorem 3 (p. 4): for any nonzero rational integer D, all solutions of x^2 + D = p^k in positive integers x, p, k with k, p > 1 satisfy k/log k < C_3 (p log p + log|D|) p log p, with C_3 an effectively computable absolute constant.
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