Prime number 31 can be expressed in the form n^5 -1, with n=2. Are there other primes that can be expressed this way?

Question

Prime number 31 can be expressed in the form n^5 -1, where n=2. Are there any other primes that can be expressed this way?

Answer 1

No other primes can be expressed as n^5-1

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n^5-1 is divisible by n-1 as per below formula. So for any value of n>2 the resulting number will be a composite number.
a^5 – b^5 = (a – b)(a^4 + a^3*b + a^2*b^2 + a*b^3 + b^4)

Answer 2

31 = [2^5 - 1] is a Mersenne Prime.
The next in the list is [2^7 - 1] = [128 - 1] = 127
[2^77232917 - 1] is the largest Mersenne Prime as of last year.