质数与合数设n=4^545+545^4,则n为质数或合数?证明
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质数与合数
设n=4^545+545^4,则n为质数或合数?证明
设n=4^545+545^4,则n为质数或合数?证明
![质数与合数设n=4^545+545^4,则n为质数或合数?证明](/uploads/image/z/18598307-59-7.jpg?t=%E8%B4%A8%E6%95%B0%E4%B8%8E%E5%90%88%E6%95%B0%E8%AE%BEn%3D4%5E545%2B545%5E4%2C%E5%88%99n%E4%B8%BA%E8%B4%A8%E6%95%B0%E6%88%96%E5%90%88%E6%95%B0%3F%E8%AF%81%E6%98%8E)
原问题是以下推广问题当2a+1=545时的特例
题:n=4^(2a+1)+(2a+1)^4,a为正整数.证明n为合数
证:
记x=2^(2a+1),y=(2a+1)^2,则2xy=(2^(a+1)*(2a+1))^2记作z^2.
n=xx+yy=(x+y)^2-2xy=(x+y+z)(x+y-z)
得证.
下面是a,n,x+y=z的值的列表:
1 145 5
2 1649 17
3 18785 65
4 268705 305
5 4208945 1465
6 67137425 6697
7 1073792449 29153
8 17179952705 122657
9 274878037265 505193
10 4398046705585 2054585
11 70368744457505 8294929
12 1125899907233250 33350257
13 18014398510013400 133776089
14 288230376152419000 535921481
15 4611686018428310000 2145452993
16 73786976294839400000 8585610305
17 1180591620717410000000 34350564553
18 18889465931478600000000 137419556185
19 302231454903657000000000 549714920945
20 4835703278458520000000000 2198937274001
21 77371252455336300000000000 8795912668985
22 1237940039285380000000000000 35183994603497
23 19807040628566100000000000000 140736699828385
24 316912650057057000000000000000 562948309256545
题:n=4^(2a+1)+(2a+1)^4,a为正整数.证明n为合数
证:
记x=2^(2a+1),y=(2a+1)^2,则2xy=(2^(a+1)*(2a+1))^2记作z^2.
n=xx+yy=(x+y)^2-2xy=(x+y+z)(x+y-z)
得证.
下面是a,n,x+y=z的值的列表:
1 145 5
2 1649 17
3 18785 65
4 268705 305
5 4208945 1465
6 67137425 6697
7 1073792449 29153
8 17179952705 122657
9 274878037265 505193
10 4398046705585 2054585
11 70368744457505 8294929
12 1125899907233250 33350257
13 18014398510013400 133776089
14 288230376152419000 535921481
15 4611686018428310000 2145452993
16 73786976294839400000 8585610305
17 1180591620717410000000 34350564553
18 18889465931478600000000 137419556185
19 302231454903657000000000 549714920945
20 4835703278458520000000000 2198937274001
21 77371252455336300000000000 8795912668985
22 1237940039285380000000000000 35183994603497
23 19807040628566100000000000000 140736699828385
24 316912650057057000000000000000 562948309256545