Useful facts Indeed Finding A Mersenne Prime

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Discussion Overview

The discussion revolves around finding a Mersenne prime, specifically the expression $2^{21609d}-1$. Participants explore methods beyond simple guesswork, examining divisibility conditions and modular arithmetic related to the expression.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant questions whether there is a better method than guess and check for finding Mersenne primes.
  • Another participant suggests that if $2^{21609d}-1$ is prime, then $21609d$ cannot be divisible by 3 or 5, leading to a consideration of remaining options.
  • Several participants discuss the implications of modular arithmetic, specifically $2^{21609d} - 1 \bmod 10$ and its simplifications.
  • One participant concludes that $d$ must be 1 based on the unit digit analysis of $2^{21609d}$ and its implications for divisibility.

Areas of Agreement / Disagreement

Participants express differing views on the implications of divisibility and modular conditions, with no consensus reached on the best method for finding Mersenne primes or the validity of specific assumptions.

Contextual Notes

Participants rely on specific assumptions about divisibility and modular properties without fully resolving the implications of these assumptions. The discussion includes unresolved mathematical steps regarding simplifications and conditions for $d$.

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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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