"Reciprocity law"의 두 판 사이의 차이

2010년 5월 14일 (금) 05:01 판

Sums of sqaures of integers 126p
equation
number of solutions of \(x^4-2x^2+2=0\) in F_p = \(1+(\frac{-1}{p})+a_p\) where
\(q\prod_{n=1}^{\infty} (1-q^{2n})(1-q^{16n})=\sum_{n=1}^\infty a_nq^n\)

[[4909919|]]

@@ 19번째 줄: / 19번째 줄: @@
 * Sums of sqaures of integers 126p
-*  equation<br> number of solutions of <math>x^4-2x^2+2=0</math> in F_p = <math>1+(\frac{-1}{p})+a_p</math> where<br><math>q\prod_{n=1}^{\infty} (1-q^{2n})(1-q^{16n})=\sum_{n=1}^\infty a_nq^n</math><br>  <br>
+*  equation<br> number of solutions of <math>x^4-2x^2+2=0</math> in F_p = <math>1+(\frac{-1}{p})+a_p</math> where<br><math>q\prod_{n=1}^{\infty} (1-q^{2n})(1-q^{16n})=\sum_{n=1}^\infty a_nq^n</math><br>
+#  Clear[g, p, M, a]<br> (*table of primes*)<br> Pr := Table[Prime[n], {n, 1, 20}]<br> (*equation*)<br> g[x_] := x^4 - 2 x^2 + 2<br> (*factorization of the discriminant & bad primes*)<br> FactorInteger[Discriminant[g[x], x]]<br> (* M[p] = number of solutions  for the equation g[x]=0 modulo p*)<br> M[n_] := 0<br> Do[For[i = 0, i < p, i++,<br>   M[p] = M[p] + If[Mod[PolynomialMod[g[i], p], p] == 0, 1, 0]], {p,<br>   Pr}]<br> (*modification of the number of solutions *)<br> a[p_] := 1 + JacobiSymbol[-1, p] + M[p]<br> (*modular form*)<br> f[q_] := Series[<br>   q*Product[(1 - q^(2 n))*(1 - q^(16 n)), {n, 1, 200}], {q, 0, 100}]<br> (*the coefficients of modular form f[q]*)<br> n[p_] := SeriesCoefficient[f[q], p]<br> (* output *)<br> title := {M_p, a_p, c_p};<br> TableForm[Table[{M[p], a[p], n[p]}, {p, Pr}] ,<br>  TableHeadings -> {Pr, title}]<br>
@@ 44번째 줄: / 46번째 줄: @@
 * [[mathematics of x^3-x+1=0]]<br>
-*   <br>
+* [[4817997|Taniyama-Shimura]]<br>