Show that for any positive integer , there exist a prime number and another positive integer such that:

(1) is a prime numer of form ;

(2) is not a multiple of ;

(3) is a multiple of .

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- September 10th 2010, 02:11 AMShankscongruence
Show that for any positive integer , there exist a prime number and another positive integer such that:

(1) is a prime numer of form ;

(2) is not a multiple of ;

(3) is a multiple of . - September 10th 2010, 05:18 AMundefined
Edit: nevermind, didn't think it through.

Edit 2: Here are some thoughts.

p being congruent to 5 mod 6 is the same as saying p is an odd prime congruent to 2 mod 3.

Here is an experiment: Note that

produces all equivalence classes mod 11 besides zero.

If this is true for all odd primes congruent to 2 mod 3 and we can prove it, then the only other piece we'd need is that there exist an infinitude of such primes. - September 10th 2010, 07:00 AMroninpro
We can rephrase this in the language of congruences:

(1)

(2)

(3) has a solution

With this in mind, it suffices to show that the map defined by is a bijection. Can you do this?