Finding Formula for Number of QN*QN Pairs Between 1 and p-1

[Ad Infinitum NAU]

I'm stumped as to how to find a formula for the number of pairs QN*QN between 1 and p-1. I must use the following as the base...

[sum]x=0 to x=p-1 of [1+(x/p)]*[1+((k-x)/p)] where the (x/p) and ((k-x)/p) terms are legendre symbols.

I also know (and can use) the fact that the number of pairs of QR (QRQR) = (1/4)*[p-4-(-1/p)] where again, (-1/p) is a legendre symbol.

Any help is greatly appreciated!

Casey

 

[quantumdude]

This will never get answered in Homework Help.

Off to Math with ye.

*kick*

 

[Hurkyl]

Refresh my memory, what do QN and QR mean?

Hurkyl

 

[Ad Infinitum NAU]

Thanks for the kick Tom..
Hurkyl,
QR represents squares mod p, QN represents non-squares mod p.

 

[Hurkyl]

I guess I should ask what "pairs" you're looking for as well; I presume you're not simply just looking for the number of ways to choose 2 elements out of QN.

 

[Ad Infinitum NAU]

Say I wrote out the numbers from 1 to p, and I underline the numbers that are QN to p. There will be "pairs" so to speak of QN within this group. By pairs I simply mean two QN lying next to each other.

 

[Hurkyl]

Aha, ok the question makes sense now.

My initial thoughts on how to solve the problem appear totally different than what you say is required. (basically a counting argument using the list of numbers 1 to p - 1)

One last question on the problem (I hope); what is k?

I was trying to figure out what the sum means; I can see how you could compute the number of pairs with the sum:

&Sigma_{x = 1 .. p - 2} (1 - (x \ p)) (1 - (x + 1 \ p)) / 4

(Where I've used the backslash to represent the Legendre symbol)

Whereas that sum you say must be used in your solution seems to be counting the number of x's such that both x and k - x are quadratic residues. It's not clear to me how that could be useful.

 

[Ad Infinitum NAU]

1. k is the constant used in the sum of (x(x^2+k)\p), where the x's are summed. (to find S(k)). I ran through it once more after a day of not thinking about it and wound up with the number of pairs of QN in p = (1/4)*(p-2+(-1\p))...

I found this by using (1/4)*sum of x from 0 to p-2 of..
((1-(x\p))*(1-((x+1)\p)-(1-(0\p))*(1-(1\p))-(1-(-1\p))*(1-(p\p))

and just running this through, cancelling and wound up with the above eqn. I'm pretty sure this is right, appreciate a verification if you could. Thanks for all the help!
casey