numbers

numbers of pair(s) of prime(p,q) satisfying the condition p² - 6q² = 1 is
A.1
B.2
C.3
D.infinite

show me step by step the solution

3 Answers

341

Hari Shankar ·2010-03-28 23:22:32

If q =2, then p=5 is a solution.

If q is an odd prime then let p = 6mÂ±1

Then 36m^2 \pm 12m = 6q^2 \Rightarrow 2m(3m \pm 2) = q^2 which means q is even, a contradiction.

So (p,q) = (5,2) is the unique solution.

edited: as pointed out by anant sir. also apologies as I didnt notice this was in 9th/10th forum.picked it up from home page

The second last line should read: "... q is even"

Imagining none to be evn, we can just write p² as 4k+1, so that we are left with
3q²=2k, which directly gives q=2.

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