Question

if P is prime then for what values of P expression 4P+1 yeilds a perfect square?

Answer 1

the number whose square it is, must be an odd number
\[4P=a^2-1\\
P=(a/2)^2-(1/4)\]
1. "a" is odd
2. P = even ?
only for P=2?

Answer 2

\[P=\left({a\over2}+{1\over2}\right)\left({a\over2}-{1\over2}\right)\]
P is a mean of two odds.
which makes it even

Answer 3

@wio please verify

Answer 4

Thats Good. Answer is 2..Thanks

Answer 5

That looks correct. You could have kept the 4 factored out though.

Answer 6

Though I'm not sure what they want when they say 'what values of P'

Answer 7

@hashim12atd how about some appreciation with a medal then?

Answer 8

@wio know.. but the steps tell me that P must be even!

Answer 9

how so?

Answer 10

ok so we are ok till "a=odd"?

Answer 11

Yeah

Answer 12

\[P=(a/2+1/2)(a/2−1/2) \] one of these factors is inevitably even . So P is even. or P=2. a= even gets us no where, as it doesn't yield
an integer

Answer 13

He said \(a\) is odd. Replace \(a\) with \(2n+1\)

Answer 14

so, from the third relation,
we conclude, (odd/2+0.5) = some number
and (odd/2-0.5) = another number should always give us an even

Answer 15

@hashim12atd and @wio me and my twisted schemas..

Answer 16

@electrokid Yeah whenever you claim something is even or odd, then use the \(2n\) and\(2n+1\) paradigm to sell it. I knew before hand even times even is even and odd times odd is odd, but that isn't always obvious.

Answer 17

how we award medals?

Answer 18

click best response.

Answer 19

@wio i know, i was trying a couple of ways and doing things mentally and lost track of penning them down

Answer 20

okay man..

Answer 21

where from are you?
electrokid

Answer 22

Pakistan.