An Elementary Solution of Lucas′ Problem

On the Diophantine equation $1^k+2^k+\dotsb+x^k=y^n$

“…Subsequent attempts by Lucas [Luc77] and Moret-Blanc [Mor76] to resolve this suffer from various defects, and it was not until a number of years later that Watson [Wat18] was able to correctly answer the question, in the negative. For more recent proofs, from a variety of perspectives, the reader is directed to [Ang90, Cuc93,Lju52,Ma85].…”

On the equation $1^k+2^k+\cdots+x^k=y^n$

“…For more recent and elementary proofs we refer to [23] and [1], [15], respectively. For a survey on this equation, see [4].…”

Lucas' square pyramid problem revisited

“…A second, more algebraic proof was found in 1952 by Ljunggren [23], though it also is somewhat on the complicated side. Attempts to repair this perceived defect have, in recent years, resulted in a number of elementary proofs, by Ma [26] and [27], Cao and Yu [6], Cucurezeanu [10] and Anglin [2]. Various generalizations, distinct from that considered here, have been addressed in [12] and [33].…”