Properties of Mersenne numbers
In Tony Reix's Properties of Mersenne and Fermat numbers online paper
you see: Mq is a prime if and only if there exists only one pair (x, y) such that: Mq = (2x)^2+ 3(3y)^2. The proof is missing. Can anybody provide a proof? By numerical testing different Mq values I have found that if Mq is composite there is no pair (x,y) that satisfies the condition. Is it possible that if Mq is composite there can be 2 or more pairs? Thanx in advance... 
[QUOTE=kurtulmehtap;241966]In Tony Reix's Properties of Mersenne and Fermat numbers online paper
you see: Mq is a prime if and only if there exists only one pair (x, y) such that: Mq = (2x)^2+ 3(3y)^2. The proof is missing. Can anybody provide a proof? [/QUOTE] I have not verified that the result is true. I will assume that it is. I will sketch a proof. This result has very little to do with Mersenne primes. Let Q = (2x)^2 + 3(3y)^2. Q is prime iff this representation is unique. Now, follow the (standard!) proof that an integer that is 1 mod 4 is prime iff it is the sum of two squares in a unique way. i.e. Factor Q over Q(sqrt(3)) and observe that you are doing so in a UFD. [QUOTE] 
[QUOTE=R.D. Silverman;241969]I have not verified that the result is true. I will assume that it is.
I will sketch a proof. This result has very little to do with Mersenne primes. Let Q = (2x)^2 + 3(3y)^2. Q is prime iff this representation is unique. Now, follow the (standard!) proof that an integer that is 1 mod 4 is prime iff it is the sum of two squares in a unique way. i.e. Factor Q over Q(sqrt(3)) and observe that you are doing so in a UFD. [/QUOTE] Pretty easy, huh? Good job. 
[QUOTE=R.D. Silverman;241969]I have not verified that the result is true. I will assume that it is.
I will sketch a proof. This result has very little to do with Mersenne primes. Let Q = (2x)^2 + 3(3y)^2. Q is prime iff this representation is unique. Now, follow the (standard!) proof that an integer that is 1 mod 4 is prime iff it is the sum of two squares in a unique way. i.e. Factor Q over Q(sqrt(3)) and observe that you are doing so in a UFD. [/QUOTE] Thanks a lot. I'm stuck. If there is a unique pair (x,y) then Q is prime,however , if Q is composite, then can we assume that there are no (x,y) pairs or should we consider there are 2,3 or more pairs? Thanks 
[QUOTE=kurtulmehtap;242188][QUOTE=R.D. Silverman;241969]I have not verified that the result is true. I will assume that it is.
I will sketch a proof. This result has very little to do with Mersenne primes. Let Q = (2x)^2 + 3(3y)^2. Q is prime iff this representation is unique. Now, follow the (standard!) proof that an integer that is 1 mod 4 is prime iff it is the sum of two squares in a unique way. i.e. Factor Q over Q(sqrt(3)) and observe that you are doing so in a UFD. Thanks a lot. I'm stuck. If there is a unique pair (x,y) then Q is prime,however , if Q is composite, then can we assume that there are no (x,y) pairs or should we consider there are 2,3 or more pairs? Thanks[/QUOTE] Hint: Composition of quadratic forms... 
[QUOTE=R.D. Silverman;242204]Hint: Composition of quadratic forms...[/QUOTE]
Another hint: think third degree polynomial equations over Z. 
So is the OPer satisfied?

[QUOTE=davar55;243127]So is the OPer satisfied?[/QUOTE]
Not Really, I am still not sure if a composite Mersenne number can have more than 1 pair for x^2 + 27y^2. There is a thesis on this subject: Mersenne primes of the form x^2+dy^2 by Bas Jansen at [url]www.math.leidenuniv.nl/en/theses/31/[/url] It has an entire section for the needed case d=27, but I still can't find the answer.. Please help. I know that I am embarassing myself but I need the answer. 
[QUOTE=kurtulmehtap;244684]Not Really, I am still not sure if a composite Mersenne number can have more than 1 pair for x^2 + 27y^2.
There is a thesis on this subject: Mersenne primes of the form x^2+dy^2 by Bas Jansen at [url]www.math.leidenuniv.nl/en/theses/31/[/url] It has an entire section for the needed case d=27, but I still can't find the answer.. Please help. I know that I am embarassing myself but I need the answer.[/QUOTE] If the number is composite, there will be more than 1. 
[QUOTE=R.D. Silverman;244687]If the number is composite, there will be more than 1.[/QUOTE]
Look up "idoneal". 
[QUOTE=R.D. Silverman;244688]Look up "idoneal".[/QUOTE]
[url]http://www.google.ca/search?sourceid=chrome&ie=UTF8&q=define:idoneal[/url] 
All times are UTC. The time now is 10:49. 
Powered by vBulletin® Version 3.8.11
Copyright ©2000  2022, Jelsoft Enterprises Ltd.