kTH POWER RESIDUE CHAINS OF GLOBAL FIELDS

In 1974, Vegh proved that if k is a prime and m a positive integer, there is an m term permutation chain of kth power residue for infinitely many primes [E.Vegh, kth power residue chains, J.Number Theory, 9(1977), 179-181]. In fact, his proof showed that 1, 2, 22, ..., 2m−1 is an m term permutation chain of kth power residue for infinitely many primes. In this paper, we prove that for k being an arbitrary positive integer, if r1, r2, ..., rm is a sequence of integers such that all sums of elements of subsets of {r1, r2, ..., rm} are distinct, then there are infinitely many primes p making it an m term permutation chain of kth power residue modulo p. It should be noted that the condition on r1, r2, ..., rm is necessary for it to be an m term permutation chain of kth power residue. From our result, we see that Vegh’s theorem holds for any positive integer k, not only for prime numbers. In fact, we prove our result in more generality where the integer ring Z is replaced by any S -integer ring of global fields (algebraic number field or algebraic function field with a finite constant field). Our main tool is Chebotarev’s density theorem for global fields.