is there only one Mersenne prime that ends in 3?

ixfd64 · 2005-11-23, 09:11

Sorry for the dumb queston.

Is 2²-1 the only Mersenne prime that ends in 3? Is it impossible for another Mersenne prime to end in 3?

Just curious.

axn · 2005-11-23, 09:16

Yes and Yes.

ixfd64 · 2005-11-23, 09:26

Thanks.

Mystwalker · 2005-11-23, 10:20

(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.

ewmayer · 2005-11-23, 19:40

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

Wacky · 2005-11-23, 20:38

Quote:

Originally Posted by ewmayer

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

How about the third Mersenne Prime?

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

xilman · 2005-11-23, 21:08

Quote:

Originally Posted by Wacky

How about the third Mersenne Prime?

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

And, of course, all Mersenne primes beyond the second. Just choose the base appropriately.

Paul

ewmayer · 2005-11-24, 02:10

Quote:

Originally Posted by Wacky

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

By common convention, unless specified otherwise the base is presumed to be 10, Mr. Smartypants. ;)

Maybe we can get the crowd over at http://www.primegrid.com 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...

Wacky · 2005-11-24, 02:41

Quote:

Originally Posted by ewmayer

Maybe we can get the crowd over at http://www.primegrid.com 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...

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. ;)

ewmayer · 2005-11-24, 19:33

Quote:

Originally Posted by 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. ;)

I thought I'd give them something a bit easier, by way of warm-up. :P

Greenbank · 2005-11-25, 12:07

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.

2005-11-23, 09:11	#1
ixfd64 Bemusing Prompter "Danny" Dec 2002 California 2³·313 Posts	is there only one Mersenne prime that ends in 3? Sorry for the dumb queston. Is 2²-1 the only Mersenne prime that ends in 3? Is it impossible for another Mersenne prime to end in 3? Just curious.

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2005-11-23, 09:16	#2
axn Jun 2003 2³×683 Posts	Yes and Yes.

2005-11-23, 09:26	#3
ixfd64 Bemusing Prompter "Danny" Dec 2002 California 2³·313 Posts	Thanks.

2005-11-23, 10:20	#4
Mystwalker Jul 2004 Potsdam, Germany 33F₁₆ Posts	(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.

2005-11-23, 19:40	#5
ewmayer ∂²ω=0 Sep 2002 República de California 26754₈ Posts	Waiting for someone to ask if there are any Mersenne primes that end in 5...

2005-11-25, 12:07	#11
Greenbank Jul 2005 2·193 Posts	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.