MHB Useful facts Indeed Finding A Mersenne Prime

[Ilikebugs]

View attachment 6518 Is there a better way than guess and check?

 

[I like Serena]

Re: Useful facts Indeed!

Is there a better way than guess and check?

Hi Ilikebugs!

If $2^{21609d}-1$ is a prime number, I think it can't be dividable by 7 or by 31.
In other words $21609d$ cannot be dividable by 3 or by 5.
As a first step, which options does that leave us? (Wondering)

 

[Ilikebugs]

Re: Useful facts Indeed!

2,4,7 and 8

 

[I like Serena]

Re: Useful facts Indeed!

2,4,7 and 8

Shouldn't that include 1?

Anyway, that leaves that:
$$2^{21609d} - 1 \bmod 10 = 7$$
Can we simplify that?

 

[Ilikebugs]

Re: Useful facts Indeed!

2^21609d mod 10 - 1 mod 10=7? I don't know

 

[Ilikebugs]

Re: Useful facts Indeed!

2^21609d-1 has a unit digit of 7 so 2^21609d has a unit digit of 8. thus 21609d is of the form 4n+3. Thus d is either 1 5 or 9 but if its 5 or 9 it would be divisible by 5 or 3. Thus d is 1

 

[I like Serena]

Re: Useful facts Indeed!

2^21609d-1 has a unit digit of 7 so 2^21609d has a unit digit of 8. thus 21609d is of the form 4n+3. Thus d is either 1 5 or 9 but if its 5 or 9 it would be divisible by 5 or 3. Thus d is 1

Good! (Nod)