A computational approach for solving $y^2=1^k+2^k+\dotsb+x^k$

Abstract

Abstract. We present a computational approach for finding all integral solutions of the equation y 2 = 1 k + 2 k + · · · + x k for even values of k. By reducing this problem to that of finding integral solutions of a certain class of quartic equations closely related to the Pell equations, we are able to apply the powerful computational machinery related to quadratic number fields. Using our approach, we determine all integral solutions for 2 ≤ k ≤ 70 assuming the Generalized Riemann Hypothesis, and for 2 ≤ k ≤ 58 unconditionally.

A fast, rigorous technique for computing the regulator of a real quadratic field

“…As observed in [9], the key to obtaining a run time complexity of O(D 1/6+ ) is that we can use subexponential techniques rather than baby steps/giant steps to compute the multiple of R. The output of the subexponential algorithms is unconditionally a multiple of R, and under the GRH is equal to R. Given this multiple, one can use the second and third stages of the O(D 1/5+ ) algorithm to determine the multiplier and thus compute R, resulting in an unconditionally correct algorithm for computing R with expected run time complexity O(D 1/6+ ) under the GRH.…”

“…Perhaps one of the most well-known examples is the problem of solving the Pell equation x 2 − Dy 2 = 1 in integers x and y for a certain nonsquare integer D, which is described in detail in Williams [25], but other examples can be found in Lenstra [15] and Jacobson et al [9]. In most cases, the fundamental unit grows at an exponential rate as the discriminant ∆ of K increases, and as a result, it is difficult to work with it explicitly.…”

“…Unfortunately, the correctness of the output of these methods also depends on the unproven GRH, and many mathematical problems that can easily be solved as soon as a good approximation of the regulator of a related real quadratic field is computed (see [9] for one such problem) require this computed value to be unconditionally correct. For example, given the regulator, one can compute a representation of the fundamental unit and use Lucas sequences to compute the coefficients of any desired unit in the field.…”

“…Up to the publication of [9], the fastest known unconditional method for computing the regulator of a real quadratic number field was the algorithm with run time complexity O(D 1/5+ ) under the GRH that is described in Lenstra [14]. The first step of the algorithm uses analytic techniques to obtain an approximation of hR, where h is the class number of the field.…”

On the Diophantine equation (x+ 1) + (x+ 2) + ... + (2x) =y

“…(x, y) = (1, 1)) solution, namely (k, n, x, y) = (2, 2, 24, 70). In 2004, Jacobson, Pintér, Walsh [10] and Bennett, Győry, Pintér [3], proved that the Schäffer's conjecture is true if 2 ≤ k ≤ 58, k is even n = 2 and 2 ≤ k ≤ 11, n is arbitrary, respectively. In 2007, Pintér [15], proved that the equation S k (x) = y 2n , in positive integers x, y, n with n > 2 (1.3) has only the trivial solution (x, y) = (1, 1) for odd values of k, with 1 ≤ k < 170.…”