Those darn paradoxes! (4): A Mathematical Brief | Tangente

In mathematics, nothing beats a good proof ----------------------------------------------

The danger of inductive reasoning based merely on amassing numerical examples—even a great many of them—is well known: a proliferation of supporting cases is no proof whatsoever and does not even increase the probability that a conjecture is true! Number theory abounds in such examples. No integer below N = 61,917,364,224 is a fifth power expressible as the sum of four fifth powers. But N, which equals 144⁵, disproves Euler's conjecture that such a decomposition was impossible; this counterexample was eventually found by computer in 1967: 144⁵ = 27⁵ + 84⁵ + 110⁵ + 133⁵.

Similarly, consider the sequence 31, 331, 3,331, 33,331, 333,331, 3,333,331, 33,333,331: they are all prime! But extrapolation is a dangerous crutch. Known since the 17^th century and revived by Simon Singh, this sequence prompted the conjecture that all numbers of this form are prime. Yet the next one is composite: 333,333,331 = 17 × 19,607,843.