The equation \(1^p+2^p+3^p+\ldots+n^p=m^q\) (Q767305)

In this paper the author shows that for fixed positive integers \(p\) and \(q\) the equation \[ S_p(n) = \sum_{i=1}^n i^p=m^g \tag{1} \] has an infinite number of solutions in natural numbers \(m\) and \(n\) only in the following cases: \(1^\circ\quad\) \(q=1\) or \(p=3\), \(q=2\), where every value of \(n\) provides a solution. \(2^\circ\quad\) \(p=1\), \(q=2\) or \(p=3\), \(q=4\) with all solutions given by \(n = \tfrac12 (x_{2r}-1)\), \(m = \tfrac12 y_{2r}\), where \(x_r+y_r\sqrt 2= (1 + \sqrt 2)^r\), \(r=1,2,3, \ldots\) \(3^\circ\quad\) \(p=5\), \(q=2\) with all solutions given by \(n=\tfrac12 (x-1)\), \(m = \tfrac14 y(3y^2+1)\), where \(x+y\sqrt 6 = (3+\sqrt 6) (5+2\sqrt 6)^s\), \(s = 0, 1, 2,\dots\,\). In all other cases the number of solutions \(N(p,q)\) of (1) is finite and \(N(p,q)\le N_0(p)\), where \(N_0(p)\) is a function of \(p\) alone. It is conjectured that the only other solutions have \(m = n = 1\), apart from \(p = q = 2\), \(n = 24\) and \(m = 70\). This conjecture is verified for a large number of values of \(p\) and \(q\). It is shown that the problem of solving (1) may be reduced to the problem of solving a finite number of equations of the form \[ | Ax^q - By^q| = 1, \tag{2} \] where \(A\) and \(B\) are bounded by \(p\) alone. By a theorem of Nagell, Ljunggren and \textit {Y. Domar} [Math. Scand. 2, 29--32 (1954; Zbl 0056.03602)] each such equation has at most two solutions in positive integers \(x\) and \(y\) for \(q\ge 3\). The equation (2) is further shown insoluble for wide classes of values of \(p\) and \(q\), where the author now makes use of a large number of known theorems concerning various diophantine equations. The reviewer remarks that the proof of Lucas for the case \(p = q = 2\) is insufficient. The first complete solution was given by \textit{G. N. Watson} [Mess. Math. 48 (1918--1919), 1--22 (1919; JFM 46.0213.01)].