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-   -   Numbers that can be written in two different ways (https://www.mersenneforum.org/showthread.php?t=25290)

 enzocreti 2020-02-24 13:21

Numbers that can be written in two different ways

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 2020-02-24 13:29

Don't forget 4

 enzocreti 2020-02-24 13:56

...primes...

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 2020-02-24 17:59

[QUOTE=enzocreti;538250]
Are there other numbers N that can be written as (2^n-4) and as (3*s^2+1)?[/QUOTE]
Quick piece of Mathematica code to test this:
[CODE]For[n = 1, n <= 10000, n++,
If[IntegerQ[Sqrt[1/3*(2^n - 5)]], Print[n, " ", 2^n - 4]]][/CODE]This is equivalent since
[CODE]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))[/CODE] Up to n = 20000 there are 3 n's such that 2^n-4 = 3s^2+1 for some integer s:
[CODE]3 (corresponding to 4)
5 (28)
9 (508)[/CODE]

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