Primes equation

[dorna]

Find the prime numbers p>0
p+7=2x^2
p^2+7=2y^2 where x,y are integers

I noticed the last digit of p is 1 and it looks like 11 it is a solution. p=11, x=3, y=8
I don't how to solve it properly though.
p(p-1)=2(x-y)(x+y) but I don't think it's going anywhere.

Any hints, please?
Thank you!!

 

[pka]

You seem to have a good grasp of what need to be done here.
Here are two comments: Every prime number is greater than [imath]1[/imath].
Moreover, every prime other than two is an odd integer. Therefore if [imath]p\ne 2[/imath] then [imath]p^2[/imath] is odd.
The sum of two odd integers is an even integer.
Frankly, I don't know, other than what you have done, what is required to do?
Can you tell us the exact wording of the question?

 

[blamocur]

Don't know how to solve this, but a quick and dirty script did not find anything besides 11 among primes under [imath]10^7[/imath].

 

[topsquark]

I don't know how to solve it either, but p being prime isn't such a big deal here. I only get one other solution (for y < 400) that gives p = 65. I suspect p being prime isn't the deal breaker for this system.

-Dan

 

[BigBeachBanana]

If you subtract the 2 equations and factor, you'll get:
[imath]p^2-p=2y^2-2x^2\implies p(p-1)=2(y-x)(y+x)[/imath]
I think you can solve this by equating [imath]p[/imath] to the term(s) in the RHS.

PS: I haven't solved it yet.

 

[blamocur]

I don't know how to solve it either, but p being prime isn't such a big deal here. I only get one other solution (for y < 400) that gives p = 65. I suspect p being prime isn't the deal breaker for this system.

-Dan

Interesting: 1, 11 and 65 are the only solutions for [imath]p<10^7[/imath].

 

[Cubist]

p(p-1)=2(x-y)(x+y) but I don't think it's going anywhere.

This introduces lots of extra solutions that don't satisfy the original 2 equations, here's some examples:-
7*(7 - 1) = 2*(11 - 10)*(11 + 10)
7*(7 - 1) = 2*(5 - 2)*(5 + 2)
17*(17 - 1) = 2*(19 - 15)*(19 + 15)
17*(17 - 1) = 2*(35 - 33)*(35 + 33)
19*(19 - 1) = 2*(14 - 5)*(14 + 5)
19*(19 - 1) = 2*(30 - 27)*(30 + 27)
19*(19 - 1) = 2*(86 - 85)*(86 + 85)
--

Instead, I recommend that you try eliminating p from the original equations.