A One-Seventh Power Problem (Posted on 2006-04-05)

	Submitted by K Sengupta
Rating: 3.3333 (3 votes)
Solution:	(Hide)
The required integer(P) is equal to 52523350145. ANALYTICAL SOLUTION (Based Upon The Solution Submitted By Tristan) In terms of provisions of the problem, it is observed that, P must necessarily possess the form n^7 + 1, where n is a positive integer. Now, in terms of provisions of the problem under reference: P^(1/7)- n < 10^-10 or,(n^7+1)^(1/7) - n < 10^-10 or,n^7+1 < (n + 10^-10)^7 or,n^7+1 < n^7 + 7n^6 * 10^-10 + ... + 7n * 10^-60 + 10^-70 or,1 < 7n^6 * 10^-10 + 21n^5 * 10^-20 + ... + 7n * 10^-60 + 10^-70 < 7n^6 * 10^-10 The magnitude of each of the terms with the exception of the first term is negligible. Accordingly, these terms (except the first term) are removed forthwith. Hence: 1 < 7n^6 * 10^-10 or,10^10 < 7n^6 or, n = ceiling( (10^10 /7)^(1/6) ) = 34 Therefore P = 34^7 + 1 = 52523350145.

	Subject	Author	Date
	Solution Acknowledgement	K Sengupta	2006-05-03 12:11:26
	Analytical solution	Tristan	2006-04-05 20:59:47
	using Excel	Bob Smith	2006-04-05 14:51:36
	using Excel	Bob Smith	2006-04-05 14:44:59