Solving the 65050-Digit Prime Puzzle

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Homework Help Overview

The problem involves demonstrating the existence of another prime number that shares the same last 65050 digits as a specified large prime, which is derived from a Mersenne prime. The context is rooted in number theory and prime distribution, particularly referencing Dirichlet's theorem on arithmetic progressions.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the implications of Dirichlet's theorem and its application to the problem, questioning how it relates to the existence of primes in an arithmetic progression. There are inquiries about the properties of Mersenne primes and the conditions for primes in sequences.

Discussion Status

Participants are actively engaging with the concepts, exploring the relationship between the given prime and the arithmetic progression. Some have proposed specific values for the arithmetic sequence and are verifying the conditions for the application of Dirichlet's theorem, while others are seeking clarification on the reasoning behind their assumptions.

Contextual Notes

There is a focus on the conditions under which the numbers in the arithmetic progression can be prime, with discussions around the coprimality of the chosen parameters. The original poster expresses a desire to learn more about the underlying mathematics rather than simply seeking a solution.

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Homework Statement



Largest known prime is the no... P = 2216091-1
consisting of 65050 digits. Show that there exists another prime that ends in the same 65050 digits as P.

Homework Equations



none

The Attempt at a Solution



Sorry, but I have totally no ideas for this one. Comes from a Math Olympiad...
Any help, appreciated!
 
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Can you elaborate on the question? What does it mean that another prime ends in the same digits?
 


The legend said:

Homework Statement



Largest known prime is the no... P = 2216091-1
consisting of 65050 digits. Show that there exists another prime that ends in the same 65050 digits as P.


Homework Equations



none

The Attempt at a Solution



Sorry, but I have totally no ideas for this one. Comes from a Math Olympiad...
Any help, appreciated!

Try looking at Dirichlet's theorem on arithmetic progressions.
 


Dick said:
Try looking at Dirichlet's theorem on arithmetic progressions.

Dirichlet's theorem works? (thought it only states that there will only be infinitely many primes number and nothing to do with the number of primes?)
Care to elaborate more? I am curious to know

There any special properties on mersenne's primes?
 
Last edited:


icystrike said:
Dirichlet's theorem works?
Care to elaborate more? I am curious to know

There any special properties on mersenne's primes?

Nothing special about Mersenne primes. The numbers larger than the given prime sharing the last 65050 digits form an arithmetic progression. Show the conditions are satisfied to assure it contains an infinite number of primes.
 


I guess it is just a special case and the statement is true for every prime.
 


Borek said:
I guess it is just a special case and the statement is true for every prime.

Except 2 and 5.
 


by any chance can you please show me how to do it?. This isn't homework nor coursework. I am just trying to learn math and some question that came up was this.

I did go through the Dirichlet's theorem(which i didnt know), in which the weaker form says...
say 2 co-prime numbers a and d are taken, then there are infinite primes in the series

a, a+d, a+2d, a+3d.....

so, this is definitely an AP. But something also states that there need not be, that the prime numbers in an arithmetic sequence are consecutive.

So, how do i proceed?


Sorry, but i don't really understand how to...
 


Then, with a bit more help...

i took d=1065050
and a is the number given.

So there should be infinite primes in that series, and all terms of that series have their numbers ending with a(the given prime).

So is this right?
 
  • #10


The legend said:
Then, with a bit more help...

i took d=1065050
and a is the number given.

So there should be infinite primes in that series, and all terms of that series have their numbers ending with a(the given prime).

So is this right?

Yes. a+n*d contains an infinite number of primes if a and d are relatively prime. Are they?
 
  • #11


yes they are.
i mean a and d are.
 
  • #12


am i right?
 
  • #13


The legend said:
am i right?

What do you think? It would help if you would say why you think they are relatively prime.
 
  • #14


ok,
so a is a odd prime number, and d is a multiple of 2 and 5 none of which can be factors of a. So yes, they are relatively prime.
 
  • #15


The legend said:
ok,
so a is a odd prime number, and d is a multiple of 2 and 5 none of which can be factors of a. So yes, they are relatively prime.

Sure. That's it. So Dirichlet applies and there are an infinite number of primes in the sequence.
 
  • #16


ok! thanks a lot :approve:
 

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