An Excerpt taken from "Algorithmic Number Theory" by Erich Bach and Jefferey Shallit.
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F. Landry [1867] proved that 1133836730401, a factor of 2^75+1, is prime. After stating the result, he went on to say

"At this point we are, if not uneasy, then at least embarrassed.

Indeed, when one has succeeded in factoring a number, and has given its factors, this can be verified immediately. But it is a different matter when the methods used fail to discover any factor, and one then asserts that the number is prime. How could one then transmit to another such a totally personal conviction? Who could be convinced, without having redone all calculations, and without having understood the principles on which those calculations were used?

We understand well that our claim is valid only as an assertion, worthwhile until someone proves the contrary, or until we make known our methods and enable others to apply them.