is there only one Mersenne prime that ends in 3?
 

Sorry for the dumb queston.

Is 2[sup]2[/sup]-1 the only Mersenne prime that ends in 3? Is it impossible for another Mersenne prime to end in 3?

Just curious.

Yes and Yes.

Thanks.

(Attempt of an) Explanantion:

(1) 2^p-1 ends with a 3 iff p = 2 + 4*x with x >= 0

(2) For 2^p-1 to be prime, p has to be prime as well.
But every suitable exponent is even, as stated in (1). The only even prime is 2.

Waiting for someone to ask if there are any Mersenne primes that end in 5...

[QUOTE=ewmayer]Waiting for someone to ask if there are any Mersenne primes that end in 5...[/QUOTE]

How about the third Mersenne Prime?


You didn't specify the number base. Try the base represented by the decimal integer "26".

[QUOTE=Wacky]How about the third Mersenne Prime?


You didn't specify the number base. Try the base represented by the decimal integer "26".[/QUOTE]And, of course, all Mersenne primes beyond the second. Just choose the base appropriately.


Paul

[QUOTE=Wacky]You didn't specify the number base. Try the base represented by the decimal integer "26".[/QUOTE]
By common convention, unless specified otherwise the base is presumed to be 10, Mr. Smartypants. ;)

Maybe we can get the crowd over at [url]http://www.primegrid.com[/url] to try their hand at searching for the first-known Mersenne prime having the property that M(p) == 5 (modulo 10). *After* they've finished cracking RSA768 via brute-force trial division, naturally...

[QUOTE=ewmayer]Maybe we can get the crowd over at [url]http://www.primegrid.com[/url] to try their hand at searching for the first-known Mersenne prime having the property that M(p) == 5 (modulo 10). *After* they've finished cracking RSA768 via brute-force trial division, naturally...[/QUOTE]

Well, now that you phrase it that way, it IS a task appropriate for the primegrid folks. However, I think that you are being a bit too restrictive on them. You should also be willing to accept any "not-yet-known" M. primes which have the property. ;)

[QUOTE=Wacky]Well, now that you phrase it that way, it IS a task appropriate for the primegrid folks. However, I think that you are being a bit too restrictive on them. You should also be willing to accept any "not-yet-known" M. primes which have the property. ;)[/QUOTE]
I thought I'd give them something a bit easier, by way of warm-up. :P

I notice their core is still hideously inefficient.

I wrote AND SENT THEM a 30 line GMP program which processed 1M candidate odd values in a matter of seconds (compared to theirs which took hours). They still aren't using it.

Anyway, a speed up of almost 5 orders of magnitude still does nothing when the size of the search space is up around 10^100.