证明 n 的4次方加上4的 n 次方是一个合数

无心人

F(175)=13*29*60169*2462728138009*936334502041597006861*850051858081746205388986877*51581197274693558656893582932788565201

无心人

PARI/GP

F1(x)=x^2+2^x-x*2^((x+1)/2)
F2(x)=x^2+2^x+x*2^((x+1)/2)

factor(F1(185))
factor(F2(185))

无心人

factornn(n)={
x=factor(F1(n));
y=factor(F2(n));
lx=matsize(x);
ly=matsize(y);
z=matrix(lx[1]+ly[1],2);
for(i=1,lx[1],z[i,1]=x[i,1]);
for(i=1,lx[1],z[i,2]=x[i,2]);
for(i=1,ly[1],z[i+lx[1],1]=y[i,1]);
for(i=1,ly[1],z[i+lx[1],2]=y[i,2]);
print(z);
}
(11:44) gp > factornn(185)
[13, 1; 73, 1; 101, 1; 229, 1; 277495254009089, 1; 8051384542367740291957603673605973, 1; 49039857307708443467467106700961152018162458760579155377, 1]
(11:44) gp > factornn(195)
[229, 1; 1117, 1; 1601, 1; 125941, 1; 188095253, 1; 485614313, 1; 10659398922613651964623234009, 1; 29, 1; 39955873, 1; 43338166377463670649253244644246017902476311614269, 1]
(11:45) gp > factornn(205)
[29, 1; 884029, 1; 2005786025520370813913570199730858271357855657122294337, 1; 110069, 1; 467179836432489518550570886961779885935956463849984682213, 1]
(11:45) gp > factornn(215)
[8429, 1; 3886061, 1; 7404961, 1; 134007397, 1; 18750620573182290341, 1; 86396511642413318321, 1; 17, 1; 3717325429, 1; 3226650637060657, 1; 258236495151773116541875635119815895533, 1]
(11:45) gp > factornn(225)
[1777, 1; 2657, 1; 11420103559343520569323370270276373579116238155515835415238513, 1; 293, 1; 1993, 1; 4493301193, 1; 20549846127838707326255676354736606761320390448942301, 1]

无心人

(11:47) gp > factornn(245)
[13, 1; 41, 1; 1741, 1; 60928846690412443891733223583277805989303998113681076270489518641449, 1; 15569, 1; 730847512814542681, 1; 8878940379663136811129, 1; 559629121570831731222642287257, 1]

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这是著名的a^4+4b^4的分解
a^4+4b^4=(a^2+2b^2)^2-4a^2b^2=(a^2+2ab+2b^2)(a^2-2ab+2b^2)
如果你初中玩过奥数，这个结论是需要背诵的