Reinventing the bicycle about Mersenne prime
 

My discovery about Mersenne prime
I find there are a lot of numbers similar to Mersenne prime and Fermat number.
(a+1)^n-a^n，(a+b)^(2^n)+a^(2^n)
Looking at [url]http://weibo.com/ttarticle/p/show?id=2309404101129943270060[/url]

[QUOTE=353085177;457639]My discovery about Mersenne prime
I find there are a lot of numbers similar to Mersenne prime and Fermat number.
(a+1)^n-a^n，(a+b)^(2^n)+a^(2^n)
Looking at [url]http://weibo.com/ttarticle/p/show?id=2309404101129943270060[/url][/QUOTE]

(a+1)^2-(a^2) describes all odd numbers as it's 2*a+1 turns out the form the link has is (a+1)^2+a^2. a square can only be 0,1, or 4 mod ( remainder on division by) 8. so without caring for the order they show up we can have 0+0,0+1,0+4,1+1,1+4,4+4 = 0,1,4,2,5,0 remainders as sums of squares in general.

[QUOTE=science_man_88;457652](a+1)^2-(a^2) describes all odd numbers as it's 2*a+1 turns out the form the link has is (a+1)^2+a^2. a square can only be 0,1, or 4 mod ( remainder on division by) 8. so without caring for the order they show up we can have 0+0,0+1,0+4,1+1,1+4,4+4 = 0,1,4,2,5,0 remainders as sums of squares in general.[/QUOTE]

You're right.

anything else to think about ? oh yes only if a and b are opposite parity will (a+b)^(2^n)+a^(2^n) be odd. edit: okay not technically true if a is even then the second part is even and you need b to be odd for there to be both an odd and even part to the sum if a is odd then b being odd is allowed as that makes a+b even leading to an even +odd = odd result.

[QUOTE=science_man_88;457659]anything else to think about ? oh yes only if a and b are opposite parity will (a+b)^(2^n)+a^(2^n) be odd. edit: okay not technically true if a is even then the second part is even and you need b to be odd for there to be both an odd and even part to the sum if a is odd then b being odd is allowed as that makes a+b even leading to an even +odd = odd result.[/QUOTE]

if a is even and b is odd = odd result
if a is odd and b is even = even result

[QUOTE=353085177;457662]if a is even and b is odd = odd result
if a is odd and b is even = even result[/QUOTE]

yes well the point is unless the powers are opposite parities the sum is even ( I realise my mistake as talking about b when I should have said a+b and a have to be opposite parities).

[QUOTE=science_man_88;457666]yes well the point is unless the powers are opposite parities the sum is even ( I realise my mistake as talking about b when I should have said a+b and a have to be opposite parities).[/QUOTE]

oh

[QUOTE=353085177;457670]oh[/QUOTE]

what else have you discovered about the mersenne numbers ?

like have you read about their factors for prime exponents ?
have you figured out that you can make the LL test similar to the trial factoring done ?

etc.

I think you'll find that factors of fermat numbers ae k*2^(n+2)+1 if I remember reading correct ( it's also how the small factor I found of F10 using PARI/GP works out. I'll be back later tonight maybe.

turns out I'm doing so much work related math as well that I forgot I don't work today.

[QUOTE=science_man_88;457682]turns out I'm doing so much work related math as well that I forgot I don't work today.[/QUOTE]

I know little about factor

[QUOTE=353085177;457692]I know little about factor[/QUOTE]

[url]http://mersenneforum.org/showthread.php?t=17126[/url] may help. though you list the form of factor much can be said about the k for specific p.