Useful facts Indeed Finding A Mersenne Prime

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SUMMARY

The discussion centers on the properties of Mersenne primes, specifically the expression $2^{21609d}-1$. It concludes that for $2^{21609d}-1$ to be prime, $21609d$ must not be divisible by 3 or 5, leading to the deduction that $d$ must equal 1. The participants analyze the modular arithmetic of the expression, confirming that the unit digit of $2^{21609d}$ is 8, which supports the conclusion about the form of $21609d$.

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

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Re: Useful facts Indeed!

Ilikebugs said:
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)
 
Re: Useful facts Indeed!

2,4,7 and 8
 
Re: Useful facts Indeed!

Ilikebugs said:
2,4,7 and 8

Shouldn't that include 1?

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

2^21609d mod 10 - 1 mod 10=7? I don't know
 
Last edited:
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
 
Re: Useful facts Indeed!

Ilikebugs said:
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)
 

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