Answer to Question #98938 in Combinatorics | Number Theory for Dorothy

Question

Does there exist a positive integer n such that n^2 − 71 is evenly divisible by 107? Prove
your assertion.

Answer 1

from n=9 to n=222 the remainder of the division is as follows:

[10, 29, 50, 73, 98, 18, 47, 78, 4, 39, 76, 8, 49, 92, 30, 77, 19, 70, 16, 71, 21, 80, 34, 97, 55, 15, 84, 48, 14, 89, 59, 31, 5, 88, 66, 46, 28, 12, 105, 93, 83, 75, 69, 65, 63, 63, 65, 69, 75, 83, 93, 105, 12, 28, 46, 66, 88, 5, 31, 59, 89, 14, 48, 84, 15, 55, 97, 34, 80, 21, 71, 16, 70, 19, 77, 30, 92, 49, 8, 76, 39, 4, 78, 47, 18, 98, 73, 50, 29, 10, 100, 85, 72, 61, 52, 45, 40, 37, 36, 37, 40, 45, 52, 61, 72, 85, 100, 10, 29, 50, 73, 98, 18, 47, 78, 4, 39, 76, 8, 49, 92, 30, 77, 19, 70, 16, 71, 21, 80, 34, 97, 55, 15, 84, 48, 14, 89, 59, 31, 5, 88, 66, 46, 28, 12, 105, 93, 83, 75, 69, 65, 63, 63, 65, 69, 75, 83, 93, 105, 12, 28, 46, 66, 88, 5, 31, 59, 89, 14, 48, 84, 15, 55, 97, 34, 80, 21, 71, 16, 70, 19, 77, 30, 92, 49, 8, 76, 39, 4, 78, 47, 18, 98, 73, 50, 29, 10, 100, 85, 72, 61, 52, 45, 40, 37, 36, 37, 40, 45, 52, 61, 72, 85, 100]

from n=223 to n=436 the remainder of the division is as follows:

[10, 29, 50, 73, 98, 18, 47, 78, 4, 39, 76, 8, 49, 92, 30, 77, 19, 70, 16, 71, 21, 80, 34, 97, 55, 15, 84, 48, 14, 89, 59, 31, 5, 88, 66, 46, 28, 12, 105, 93, 83, 75, 69, 65, 63, 63, 65, 69, 75, 83, 93, 105, 12, 28, 46, 66, 88, 5, 31, 59, 89, 14, 48, 84, 15, 55, 97, 34, 80, 21, 71, 16, 70, 19, 77, 30, 92, 49, 8, 76, 39, 4, 78, 47, 18, 98, 73, 50, 29, 10, 100, 85, 72, 61, 52, 45, 40, 37, 36, 37, 40, 45, 52, 61, 72, 85, 100, 10, 29, 50, 73, 98, 18, 47, 78, 4, 39, 76, 8, 49, 92, 30, 77, 19, 70, 16, 71, 21, 80, 34, 97, 55, 15, 84, 48, 14, 89, 59, 31, 5, 88, 66, 46, 28, 12, 105, 93, 83, 75, 69, 65, 63, 63, 65, 69, 75, 83, 93, 105, 12, 28, 46, 66, 88, 5, 31, 59, 89, 14, 48, 84, 15, 55, 97, 34, 80, 21, 71, 16, 70, 19, 77, 30, 92, 49, 8, 76, 39, 4, 78, 47, 18, 98, 73, 50, 29, 10, 100, 85, 72, 61, 52, 45, 40, 37, 36, 37, 40, 45, 52, 61, 72, 85, 100]

and so on.....

They are the same, that's why the answer is NO. A positive number satisfying the property does not exist.