Numbers that can be written in two different ways

[enzocreti]

4, 28 and 508 can be written as (2^n-4) for some n, but also as:


(3*s^2+1) for some s




Are there other numbers N that can be written as (2^n-4) and as (3*s^2+1)?

[axn]

Don't forget 4

[enzocreti]

5, 29, 509 are primes






such that can be written as 3*n^2+2 and as 2^s-3 for some n and s




do you believe they are infinite?

[Dylan14]

Originally Posted by enzocreti

Are there other numbers N that can be written as (2^n-4) and as (3*s^2+1)?

Quick piece of Mathematica code to test this:

For[n = 1, n <= 10000, n++, 
 If[IntegerQ[Sqrt[1/3*(2^n - 5)]], Print[n, " ", 2^n - 4]]]

This is equivalent since

2^n-4=3s^2+1
2^n-5=3s^2
s^2=1/3*(2^n-5)
s = sqrt(1/3*(2^n-5))

Up to n = 20000 there are 3 n's such that 2^n-4 = 3s^2+1 for some integer s:

3 (corresponding to 4)
5 (28)
9 (508)