A program automatically solving diophantine equations of the form a ^ x + b = c ^ y for its positive integer solutions (x, y), where a, b and c are given positive integers satisfying a >= 2 and c >= 2.
Due to the limitations of the int type range, this program can only ensure successful solving of equations within the range of 2 <= a <= 70, b <= 30 and 2 <= c <= 70. However, there are many equations outside this range that can also be solved successfully.
Implemented in pure SysY.
./diophantineYou need to input a, b and c first.
$5^x+3=2^y,\ x,y\in\mathbb{N}^*\Rightarrow(x,y)=(1, 3)\ or\ (x,y)=(3, 7)$
Please input a (a should be a positive integer >= 2):
5
Please input b (b should be a positive integer):
3
Please input c (c should be a positive integer >= 2):
2
For positive integers x, y satisfying 5 ^ x + 3 = 2 ^ y,
if y >= 8, 5 ^ x = 253 (mod 256).
So x = 35 (mod 64),
which implies x = 35, 99, 163, 227 (mod 256).
Therefore, 5 ^ x = 14, 224, 243, 33 (mod 257).
So 2 ^ y = 17, 227, 246, 36 (mod 257), but this is impossible.
Therefore, y < 8.
Further examination shows that (x, y) = (1, 3), (3, 7).
$3^x+10=13^y,\ x,y\in\mathbb{N}^*\Rightarrow(x,y)=(1, 1)\ or\ (x,y)=(7, 3)$
Please input a (a should be a positive integer >= 2):
3
Please input b (b should be a positive integer):
10
Please input c (c should be a positive integer >= 2):
13
For positive integers x, y satisfying 3 ^ x + 10 = 13 ^ y,
if x >= 8, 13 ^ y = 10 (mod 6561).
So y = 1461 (mod 2187),
which implies y = 1461, 3648, 5835, 8022 (mod 8748).
Therefore, 13 ^ y = 11616, 6486, 5881, 11011 (mod 17497).
So 3 ^ x = 11606, 6476, 5871, 11001 (mod 17497), but this is impossible.
Therefore, x < 8.
Further examination shows that (x, y) = (1, 1), (7, 3).
$2^x+89=91^y,\ x,y\in\mathbb{N}^*\Rightarrow(x,y)=(1, 1)\ or\ (x,y)=(13, 2)$
Please input a (a should be a positive integer >= 2):
2
Please input b (b should be a positive integer):
89
Please input c (c should be a positive integer >= 2):
91
For positive integers x, y satisfying 2 ^ x + 89 = 91 ^ y,
if y >= 3, 2 ^ x = 254 (mod 343).
So x = 76 (mod 147),
which implies x = 76, 223, 370, 517, 664, 811, 958, 1105, 1252 (mod 1323).
Therefore, 2 ^ x = 1994, 852, 1811, 957, 1447, 1513, 2343, 348, 1970 (mod 2647).
So 91 ^ y = 2083, 941, 1900, 1046, 1536, 1602, 2432, 437, 2059 (mod 2647), but this is impossible.
Therefore, y < 3.
Further examination shows that (x, y) = (1, 1), (13, 2).
$x,y\in\mathbb{N}^*\Rightarrow 101^x+3\neq103^y$
Please input a (a should be a positive integer >= 2):
101
Please input b (b should be a positive integer):
3
Please input c (c should be a positive integer >= 2):
103
For positive integers x, y satisfying 101 ^ x + 3 = 103 ^ y,
if x >= 1, 103 ^ y = 3 (mod 101).
So y = 69 (mod 100),
which implies y = 19 (mod 25).
Therefore, 103 ^ y = 583 (mod 701).
So 101 ^ x = 580 (mod 701), but this is impossible.
Therefore, x < 1.
So 101 ^ x + 3 = 103 ^ y is impossible.