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Old 2020-02-24, 13:21   #1
enzocreti
 
Mar 2018

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Default 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)?

Last fiddled with by enzocreti on 2020-02-24 at 13:51
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Old 2020-02-24, 13:29   #2
axn
 
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Don't forget 4
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Old 2020-02-24, 13:56   #3
enzocreti
 
Mar 2018

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Default ...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?
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Old 2020-02-24, 17:59   #4
Dylan14
 
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Quote:
Originally Posted by enzocreti View Post
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:
Code:
For[n = 1, n <= 10000, n++, 
 If[IntegerQ[Sqrt[1/3*(2^n - 5)]], Print[n, " ", 2^n - 4]]]
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))
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)
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