A number theory problem by Ilham Saiful Fauzi

Number Theory Level 3

Find the greatest positive integer n n such that n 4 + 4 n n^{4}+4^{n} is a prime number


The answer is 1.

Ilham Saiful Fauzi by Ilham Saiful Fauzi , 25, Indonesia

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1 solution

If n is even, the number clearly divisible by 2. If n is odd, assume that n = 2 k + 1 n=2k+1 such that we can write n 4 + 4 n = ( n 2 + 2 k + 1 n + 2 k + 1 ) ( n 2 − 2 k + 1 n + 2 k + 1 ) n^{4}+4^{n}=(n^{2}+2^{k+1}n+2^{k+1})(n^{2}-2^{k+1}n+2^{k+1}) in which both the factors are greater than 1. It means that the number is composite. Thus the number is prime for n = 1 , n 4 + 4 n = 1 + 4 = 5 n=1, n^{4}+4^{n}=1+4=5

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can you explain in detail.

Sarthak Singla - 5 years, 8 months ago

We have a pattern

For n^4 is 1,6,1,6,5,6,1,6,1,0,1,6,...

For 4^n is 4,6,4,6,4,6,....

We can see that n is odd.

Ananda Sp - 5 years ago

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