Solution to Problem 75

Congratulations to this week's winner

Paul Leisher

who showed there are no solutions for n < 15,000, t < 10,000, k < 1,000. Partial solutions were also received from Nathan Pauli, Ray Kremer. Solutions were submitted by Ivan Lisac, Alan Zimmermann, William Troxel, Osama Alkam, Emanuele Macrì, Lorenzo Pozzoli, Massimo Brignone.

There are no solutions to the equation. Here's an explanation.

Note that n and n+1 differ by only 1, so can have no common prime factor. Let p₁, p₂,..., p_k be the primes dividing n, and q₁, q₂,..., q_j be the primes dividing n+1. Since each prime factor of t appears a multiple of k times, it follows that n is of the form r^kand n+1 is of the form s^k; here the primes p₁, p₂,..., p_k are the prime divisors of r, while the primes q₁, q₂,..., q_jare the prime divisors of s. But

1	= (n + 1) - n = s^k - r^k
	= (s - r)(s^k-1+ s^k-2r + s^k-3r² + s^k-4r³ + ... + s²r^k-3 + sr^k-2 + r^k-1).

Since both factors on the right-hand side are integers, and the second one is clearly positive, each must equal 1. But the second factor is clearly greater than 1 whenever k > 1.

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Page last updated 16 November 1999.