[QUOTE=davieddy;144106]Come to think of it, Mersenne's prediction of
31, 67, 127 and 257 suggests he had a bee in his bonnet
about exponents close to powers of 2.[/QUOTE]

"Fermat envy", most probably.

[quote=Mini-Geek;144076]True, so cheesehead's statement really isn't right, but I doubt that Mersenne was any better than statistical chance would have him for the Mersennes he predicted (remember how small these ones were).[/quote]
I don't think "small" is an adjective often applied to 2^31.
As for statistically expected guesses, I would say 2 out of 4
was lucky, considering the number of primes up to 257.

[quote=davieddy;144215]I don't think "small" is an adjective often applied to 2^31.[/quote]
True, but when you really look at it, we're not really talking about the scale of 2^31 or even 31 - since we're only talking about the primes in that range, we're really talking about the scale of the primes in that range. He was still quite lucky (or perhaps he eliminated small factors from some in that range then guessed to finish the rest) considering the number of primes, Mersenne primes, incorrect guesses, and correct guesses.

Did he know that factors had to be (2kp + 1)?
Even if he did he would have his work cut out proving many
of the Mersenne numbers up to 2^257-1 composite.
With only 5 Mersenne primes among all Mersenne numbers in
the range, 2 out of 4 was good guessing!

[QUOTE=davieddy;144106]Come to think of it, Mersenne's prediction of
31, 67, 127 and 257 suggests he had a bee in his bonnet
about exponents close to powers of 2.[/QUOTE]

[URL="http://mathworld.wolfram.com/NewMersennePrimeConjecture.html"]http://mathworld.wolfram.com/NewMersennePrimeConjecture.html[/URL]

Dickson states "In a letter to Tanner [L'intermediaire des math., 2, 1895, 317] Lucas stated that Mersenne (1644, 1647) implied that a necessary and sufficient condition that 2[sup]p[/sup]-1 be a prime is that be a prime of one of the forms 2[sup]2n[/sup]+1 , 2[sup]2n[/sup]+-3 , 2[sup]2n+1[/sup]-1."

[quote=davieddy;144253]Did he know that factors had to be (2kp + 1)?
[/quote]
According to
[URL]http://primes.utm.edu/mersenne/index.html[/URL]
Fermat proved 2^23-1 and 2^37-1 composite in 1640,
so he might have proved the 2kp+1 thing by then.
Mersenne's conjecture was 1644.

1640 is also the date Fermat stated his "Little Theorem".

[quote=ATH;144290][URL]http://mathworld.wolfram.com/NewMersennePrimeConjecture.html[/URL]

Dickson states "In a letter to Tanner [L'intermediaire des math., 2, 1895, 317] Lucas stated that Mersenne (1644, 1647) implied that a necessary and sufficient condition that 2[sup]p[/sup]-1 be a prime is that be a prime of one of the forms 2[sup]2n[/sup]+1 , 2[sup]2n[/sup]+-3 , 2[sup]2n+1[/sup]-1."[/quote]
How come he missed 61 then?

plausible?
 

[QUOTE=davieddy;144293]How come he missed 61 then?[/QUOTE]

Wells in the "Penguin Book of Interesting Numbers" states that 67 could have been a misprint for 61. But he doesn't give a source so this could be no more than a Wellsian musing. Its possible though.

Richard

[quote=Richard Cameron;144296]Wells in the "Penguin Book of Interesting Numbers" states that 67 could have been a misprint for 61. But he doesn't give a source so this could be no more than a Wellsian musing. Its possible though.[/quote]Wouldn't that only change the question to, "Then why didn't he guess 67 as well as, or instead of, 61, since they're both 2[sup]2n[/sup]+-3?"

An unnecessary addition that may be deleted:

[tex]2^{257}-1[/tex] was proved composite in 1922 by Kraitchik, according to the Dictionary of C and I Numbers.

[quote=10metreh;151680]An unnecessary addition that may be deleted:

[/quote]
On the contrary, it is probably your most constructive post to date.