费马小定理4
任一正整数N都可以表为不超过四个平方数的和。
即:N = a^2+b^2+c^2+d^2
拉格朗日分类证明:
4k+1型素数p=a^2+b^2
8k+3型素数p=a^2+b^2+c^2
8k+7型素数p=a^2+b^2+c^2+d^2
s = 0;
For[k = 0, k <= 100, k++, If[PrimeQ[8 k + 3], s = s + 1;
Print[s, "-----", 8 k + 3, "-----",
PowersRepresentations[8 k + 3, 3, 2]]]]
1-----3-----{{1,1,1}}
2-----11-----{{1,1,3}}
3-----19-----{{1,3,3}}
4-----43-----{{3,3,5}}
5-----59-----{{1,3,7},{3,5,5}}
6-----67-----{{3,3,7}}
7-----83-----{{1,1,9},{3,5,7}}
8-----107-----{{1,5,9},{3,7,7}}
9-----131-----{{1,3,11},{1,7,9},{5,5,9}}
10-----139-----{{3,3,11},{3,7,9}}
11-----163-----{{1,9,9}}
12-----179-----{{1,3,13},{3,7,11},{7,7,9}}
13-----211-----{{3,9,11},{7,9,9}}
14-----227-----{{1,1,15},{3,7,13},{5,9,11}}
15-----251-----{{1,5,15},{1,9,13},{3,11,11},{7,9,11}}
16-----283-----{{3,7,15},{9,9,11}}
17-----307-----{{1,9,15},{3,3,17}}
18-----331-----{{5,9,15},{9,9,13}}
19-----347-----{{1,11,15},{3,7,17},{3,13,13}}
20-----379-----{{3,3,19},{3,9,17}}
21-----419-----{{3,7,19},{3,11,17},{5,13,15},{7,9,17},{9,13,13}}
22-----443-----{{1,1,21},{1,9,19},{7,13,15}}
23-----467-----{{1,5,21},{3,13,17},{5,9,19},{11,11,15}}
24-----491-----{{1,7,21},{3,11,19},{5,5,21},{7,9,19},{9,11,17}}
25-----499-----{{3,7,21},{7,15,15}}
26-----523-----{{1,9,21},{3,15,17},{9,9,19}}
27-----547-----{{3,3,23},{5,9,21}}
28-----563-----{{1,11,21},{3,5,23},{7,15,17},{9,11,19},{13,13,15}}
29-----571-----{{3,11,21},{7,9,21},{11,15,15}}
30-----587-----{{1,15,19},{3,7,23},{3,17,17},{5,11,21}}
31-----619-----{{3,9,23},{3,13,21},{13,15,15}}
32-----643-----{{3,3,25},{9,11,21}}
33-----659-----{{3,5,25},{3,11,23},{3,17,19},{7,9,23},{7,13,21},{9,17,17}}
34-----683-----{{3,7,25},{11,11,21},{13,15,17}}
35-----691-----{{5,15,21},{9,9,23},{9,13,21}}
36-----739-----{{1,3,27},{3,17,21},{15,15,17}}
37-----787-----{{3,7,27},{9,9,25},{11,15,21}} |