x^3+y^4=z^5 的正整数解

王守恩

A300565——Numbers z such that there is a solution to x^3 + y^4 = z^5 with x, y, z >= 1.

z=32, 250, 1944, 2744, 3888, 19208, 27648, 55296, 59049, 59582, 81000, 82944, 131072, 135000, 185193, 200000,——共16项还没有我们的多！

# A300565 (b-file synthesized from sequence entry)——后面还有大把地皮空置——我来把 y,x 补上。
1 32{{y -> 64, x -> 256}}
2 250{{y -> 625, x -> 9375}}
3 1944{{y -> 11664, x -> 209952}}
4 2744{{y -> 19208, x -> 268912}}
5 3888{{y -> 23328, x -> 839808}}
6 19208{{y -> 134456, x -> 13176688}}
7 27648{{y -> 331776, x -> 15925248}}
8 55296{{y -> 331776, x -> 79626240}}
9 59049{{y -> 531441, x -> 86093442}}
10 59582{{y -> 923521, x -> 28629151}}
11 81000{{y -> 1215000,x -> 109350000}}
12 82944{{y -> 995328,  x -> 143327232}}
13 131072{{y -> 2097152, x -> 268435456}}
14 135000{{y -> 2025000, x -> 303750000}}
15 185193{{y -> 3518667, x -> 401128038}}
16 200000{{y -> 4000000, x -> 400000000}}

王守恩

猜想: 不定方程 $x^a+y^b=z^c$,  x, y, z, a, b, c = 正整数,  a > 1,   b > 1,   c > 1,  满足$GCD(LCM(a,b),c)=1$

当 \(\D x=2^{u/a},y=2^{u/b},z=2^{v}\) 时,  不定方程 $x^a+y^b=z^c$ 有最小解。

其中: \(\D u=v\cdot c-1=s\cdot LCM(a,b),\ s<c\)

Table[Solve[{z == 2^v, y == 2^(u/b), x == 2^(u/a), u == v*c - 1 == s*LCM[a, b], s < c}, {s, v, u, z, y, x}, PositiveIntegers], {a, 2, 9}, {b, a, 9}, {c, 7, 7}]

既然是猜想就是希望有人推翻她！！！

王守恩

谢谢 northwolves！2^2=1+3 可以！

Table[Solve[{z == 3^v, y == 3^(u/b), x == 2*3^(u/a), u == v*c - 2 == s*LCM[a, b], s < c}, {s, v, u, z, y, x}, PositiveIntegers], {a, 2, 9}, {b, a, 9}, {c, 7, 7}]