Find all prime numbers and all positive integers such that
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The only solution is . clearly has to be odd. If , then must also be divisible by 5; if it isn’t, would be divisible by 5 by Fermat’s little theorem. Hence if , let , . Then is a product of two integers both greater than 1 and hence cannot be prime.
We get only 1 as the solution!!!!!
Yes, that's what JaneBennet has proven. N = 1 is the only solution yielding K = 5.
Another variant of the problem is this : Find all n such that n^4+4^n is a prime!
Originally Posted by manjil Another variant of the problem is this : Find all n such that n^4+4^n is a prime! See This
Originally Posted by manjil We get only 1 as the solution!!!!! Not even wrong, a solution consists of a pair a positive integer N and a prime K. As Miss Bennet demonstrates there is indeed only one solution and it is N=1, K=5 RonL
Originally Posted by perash Find all prime numbers and all positive integers such that I found a better solution. If is even, then is an even number greater than 2, so it is not prime. If is odd, let , . Then . If , both the factors are greater than 1 and so is not prime. Hence the only solution is when , i.e. .
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