When Extrapolation Fails Us: Incorrect Mathematical Conjectures

It is well-known that our intuition is not perfect. This sort of situation seems to be embodied in a Quora question that I recently came across: What is an example of a conjecture that was proven wrong for "very large" numbers? Essentially, the questioner was interested in situations where a mathematical conjecture appeared true, but fails only with the application of advanced computational power as cases far beyond human abilities are tested. Another example is a conjecture from Euler: The 17th century Swiss mathematician Leonhard Euler claimed that there are no whole number solutions to an equation not dissimilar to Fermat’s equation: Euler’s equation: x4 + y4 + z4 = w4 For two hundred years nobody could prove Euler’s conjecture, but on the other hand nobody could disprove it by finding a counter-example. Then in 1988 Noam Elkies of Harvard University discovered the following solution: 2,682,4404 + 15,365,6394 + 18,796,7604 = 20,615,6734 Despite all the previous evidence, Euler’s conjecture turned out to be false.