Solving the 65050-Digit Prime Puzzle

[The legend]

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!

 

[chaoseverlasting]

Can you elaborate on the question? What does it mean that another prime ends in the same digits?

 

[Dick]

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.

 

[icystrike]

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?

 

[Dick]

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.

 

[Borek]

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

 

[Dick]

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

Except 2 and 5.

 

[The legend]

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...

 

[The legend]

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?

 

[Dick]

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?

 

[The legend]

yes they are.
i mean a and d are.

 

[The legend]

am i right?

 

[Dick]

am i right?

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

 

[The legend]

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.

 

[Dick]

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.

 

[The legend]

ok! thanks a lot