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Number of Solutions to d(p) 
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#1
Aug2909, 12:40 PM

P: 64

What is the number of solutions d(p) of
[tex]Nn^2 \equiv 0 \pmod p[/tex] where p is a prime and n and N are positive and N => n? 


#2
Aug2909, 01:37 PM

P: 21

I'm not sure what you are really asking, but for each prime p, there are an infinite number of solutions (N,n) satisfying your criteria. Take n = 1, and N= k*p + 1, for k = 1, 2, 3, ...



#3
Aug2909, 01:39 PM

P: 64




#4
Aug2909, 04:06 PM

P: 64

Number of Solutions to d(p)
I am an idiot.
Hint: quadratic residue 


#5
Aug2909, 05:24 PM

P: 21

I still don't understand what you are asking.
[tex]$ \{ (N,n) : N \ge 1, n \ge 1, N \ge n, N \equiv n^2 \pmod p \} [/tex] Clearly you intend something different. 


#6
Aug2909, 07:28 PM

P: 64

Let F(n) be a polynomial of degree g => 1 with integer coefficients. Let d(p) denote the number of solutions to the congruency [tex]F(n) \equiv 0 \pmod p[/tex] for all primes p (and suppose that d(p) < p for all p). We may take F(n) = Nn^2, where N is an integer greater than (or equal to) n. What is d(p) then in this case? 


#7
Aug2909, 07:46 PM

P: 21

[tex]$ \max_{N \ge 1} \left \{ n \pmod p : N \equiv n^2 \pmod p \} \right [/tex] Then d(2) = 1, d(p) = 2 for odd primes p. 


#8
Aug3109, 01:46 PM

P: 57




#9
Aug3109, 01:57 PM

P: 21

Perhaps "d(p,N) = exact number of solutions" would be a more useful function. 


#10
Aug3109, 01:59 PM

P: 57

I think you mean that N is fixed , but
If the question is : Find all N < p such that n^2 = N ( mod p) for some n ( By some n , I mean that corresponding to each N there will be one n) , then : d(p) = greatest integer less than square root of p 


#11
Aug3109, 02:11 PM

P: 57

take remainder of N divided by p . Let it be r. Now if r is a perfect square , then d(p.N) will have infinite solutions of the form r + kp else there is no solution 


#12
Sep709, 06:43 PM

P: 64

Would d(p) be different if it was instead the number of solutions to:
[tex]F(n) \equiv 0 \pmod p[/tex]? Here F(n) = n  q where q = m^2 is a perfect square where m is a natural number. 


#13
Sep809, 04:56 PM

Sci Advisor
HW Helper
P: 3,684

Is m fixed or can it take any integer value? Are there any limits on the value of n or m?



#14
Sep809, 05:43 PM

P: 64




#15
Sep909, 02:07 PM

Sci Advisor
HW Helper
P: 3,684




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