On the incongruence of consecutive fourth powers
The following problem is solved in the paper of \textit{L. K. Arnold}, \textit{S. J. Benkoski} and \textit{B. J. McCabe} [Am. Math. Mon. 92, 275-277 (1985; Zbl 0573.10002)]: Given an integer \(n\geq 1\), what is the smallest positive integer k such that \(1^2, 2^2,\ldots,n^2\) are all incongruent modulo \(k\)? Let \(D(n)\) denote this integer; they show that \(D(n)=1\) if \(n=1\); \(D(n)=2\) if \(n=2\); \(D(n)=9\) if \(n=4\); and \(D(n)=\min \{k \mid k\geq 2n\) and \(k=p\) or \(2p\) with \(p\) prime\(\}\), for all other \(n\). Here the following generalization is obtained: For \(n\geq 1\) and \(j\geq 1\), define \(D(j,n):=\min \{k\geq 1 \mid a^ j\equiv b^ j\pmod k\) if \(1\leq a<b\leq n\}\). Then \(D(2^h,n)\) for all \(n\geq 1\) and \(h\geq 2\) are determined. The authors also mention what is known about \(D(j,n)\) for other values of \(j\). [See also the subsequent review Zbl 0712.11003.]
incongruence of consecutive fourth powers
scientific article; zbMATH DE number 4170990