20090829, 05:48  #1 
Dec 2008
2·5·83 Posts 
Number of Solutions to d(p)
What is the number of solutions d(p) of
where p is a prime and n and N are positive and N => n? Last fiddled with by flouran on 20090829 at 05:56 
20090829, 10:19  #2 
Sep 2006
Brussels, Belgium
2×797 Posts 
I suppose that another condition is N<p, otherwise d(p) is infinite : any N=n^{2} satisfies Nn^{2}=0 and n <= N.
With those conditions : integers n, N and p where p is prime and 0 < n <= N < p, the solutions with n^{2} < p are trivial and their number is int(p^0,5)+1. Only the solutions like 43^{2} mod 5 are interesting. I have no answer though. Jacob 
20090829, 14:12  #3 
Dec 2008
2×5×83 Posts 
I only have a trivial estimate for d(p). That is, d(p) < p1 if .
But that's not at all interesting.... Last fiddled with by flouran on 20090829 at 14:12 
20090829, 20:40  #4 
Aug 2006
2×2,969 Posts 
This is covered in any elementary number theory textbook. Look up "quadratic residue" on Google (or, better, Ireland & Rosen).

20090829, 21:12  #5 
Dec 2008
33E_{16} Posts 
So basically, if we let p be a prime, then since the congruence has only the solution n=0, then has only 1 solution as well?
Last fiddled with by flouran on 20090829 at 21:16 
20090829, 21:17  #6 
Aug 2002
Ann Arbor, MI
433 Posts 

20090829, 22:38  #7 
Aug 2006
2·2,969 Posts 
So you think 0 < n <= N <= p was intended?
Last fiddled with by CRGreathouse on 20090829 at 22:53 
20090829, 22:54  #8 
Aug 2006
2·2,969 Posts 
I can't come up with any interpretation under which that would be true. Can you explain in more detail what you mean? The best guess I have gives 2, 2, 4, 3, 6, 8, 11, 10, 9, 18, 13, 20, ... solutions for p = 2, 3, 5, 7, 11, ....

20090830, 00:26  #9 
Dec 2008
2×5×83 Posts 
Let F(n) be a polynomial of degree g => 1 with integer coefficients. Let d(p) denote the number of solutions to the congruency 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?
Last fiddled with by flouran on 20090830 at 00:32 
20090830, 01:03  #10 
Aug 2002
Ann Arbor, MI
433_{10} Posts 

20090830, 01:23  #11 
Dec 2008
2×5×83 Posts 

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