ON QUADRUPLES OF CONSECUTIVE ¿th POWER RESIDUES

In [2] it was shown that A(&, 4) = co for fe^ 1048909 and it was conjectured that A(&, 4) = =° for all k. In this paper we establish this conjecture with the following Theorem. A(&, 4) = ». Proof. It suffices to prove the theorem for values of k which are prime. The proof makes use of the following proposition which is a special case of a result of Kummer [l] (see also [3]). Proposition. Let k be a prime and let yi, • • • , y„ be an arbitrary sequence of kth roots of unity. Then there exist infinitely many primes p with corresponding kth power character % modulo p such that xipi) = 7i, H«á », where pi denotes the ith prime. Thus, for any n and prime k, there exists a prime p with corresponding jfeth power character x modulo p such that X(2) * I, xiPi) = 1, 2ûiûn. Now consider any four consecutive positive integers all less than pn. It is clear that exactly one of these integers must equal 2(2¿ + l) for some integer d. But we have x(2(2d + 1)) = x(2)x(2d + 1) = x(2)-l ^ 1 since 2d + l is the product of odd primes less than pn. Therefore Received by the editors December 29, 1962. 196