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imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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| 19번째 줄: | 19번째 줄: | ||
* Sums of sqaures of integers 126p | * Sums of sqaures of integers 126p | ||
| − | * equation | + | * equation number of solutions of <math>x^4-2x^2+2=0</math> in F_p = <math>1+(\frac{-1}{p})+a_p</math> where<math>q\prod_{n=1}^{\infty} (1-q^{2n})(1-q^{16n})=\sum_{n=1}^\infty a_nq^n</math> |
| − | # Clear[g, p, M, a] | + | # Clear[g, p, M, a] (*table of primes*) Pr := Table[Prime[n], {n, 1, 20}] (*equation*) g[x_] := x^4 - 2 x^2 + 2 (*factorization of the discriminant & bad primes*) FactorInteger[Discriminant[g[x], x]] (* M[p] = number of solutions for the equation g[x]=0 modulo p*) M[n_] := 0 Do[For[i = 0, i < p, i++, M[p] = M[p] + If[Mod[PolynomialMod[g[i], p], p] == 0, 1, 0]], {p, Pr}] (*modification of the number of solutions *) a[p_] := 1 + JacobiSymbol[-1, p] + M[p] (*modular form*) f[q_] := Series[ q*Product[(1 - q^(2 n))*(1 - q^(16 n)), {n, 1, 200}], {q, 0, 100}] (*the coefficients of modular form f[q]*) n[p_] := SeriesCoefficient[f[q], p] (* output *) title := {M_p, a_p, c_p}; TableForm[Table[{M[p], a[p], n[p]}, {p, Pr}] , TableHeadings -> {Pr, title}] |
| 45번째 줄: | 45번째 줄: | ||
==related items== | ==related items== | ||
| − | * [[mathematics of x^3-x+1=0]] | + | * [[mathematics of x^3-x+1=0]] |
| − | * [[4817997|Taniyama-Shimura]] | + | * [[4817997|Taniyama-Shimura]] |
| 66번째 줄: | 66번째 줄: | ||
| − | * [[2010년 books and articles]] | + | * [[2010년 books and articles]] |
* http://gigapedia.info/1/squares | * http://gigapedia.info/1/squares | ||
* http://gigapedia.info/1/ | * http://gigapedia.info/1/ | ||
| 103번째 줄: | 103번째 줄: | ||
==blogs== | ==blogs== | ||
| − | * 구글 블로그 검색 | + | * 구글 블로그 검색 |
** http://blogsearch.google.com/blogsearch?q= | ** http://blogsearch.google.com/blogsearch?q= | ||
** http://blogsearch.google.com/blogsearch?q= | ** http://blogsearch.google.com/blogsearch?q= | ||
2020년 11월 13일 (금) 02:25 판
introduction
example 1
- Diamond & Shurman 155p
- \(x^3=d\)
example 2
- 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\(q\prod_{n=1}^{\infty} (1-q^{2n})(1-q^{16n})=\sum_{n=1}^\infty a_nq^n\)
- Clear[g, p, M, a] (*table of primes*) Pr := Table[Prime[n], {n, 1, 20}] (*equation*) g[x_] := x^4 - 2 x^2 + 2 (*factorization of the discriminant & bad primes*) FactorInteger[Discriminant[g[x], x]] (* M[p] = number of solutions for the equation g[x]=0 modulo p*) M[n_] := 0 Do[For[i = 0, i < p, i++, M[p] = M[p] + If[Mod[PolynomialMod[g[i], p], p] == 0, 1, 0]], {p, Pr}] (*modification of the number of solutions *) a[p_] := 1 + JacobiSymbol[-1, p] + M[p] (*modular form*) f[q_] := Series[ q*Product[(1 - q^(2 n))*(1 - q^(16 n)), {n, 1, 200}], {q, 0, 100}] (*the coefficients of modular form f[q]*) n[p_] := SeriesCoefficient[f[q], p] (* output *) title := {M_p, a_p, c_p}; TableForm[Table[{M[p], a[p], n[p]}, {p, Pr}] , TableHeadings -> {Pr, title}]
example 3
- 1-2-3- of modular forms
history
encyclopedia
books
- 2010년 books and articles
- http://gigapedia.info/1/squares
- http://gigapedia.info/1/
- http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
articles
- http://www.ams.org/mathscinet
- [1]http://www.zentralblatt-math.org/zmath/en/
- [2]http://arxiv.org/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html
- http://dx.doi.org/
question and answers(Math Overflow)
blogs
experts on the field