Teenager Solves Stubborn Riddle About Prime Number Look-Alikes
Fnord666 writes:
Teenager Solves Stubborn Riddle About Prime Number Look-Alikes
Over a century ago, in that quest for a fast, powerful primality test, mathematicians stumbled on a group of troublemakers - numbers that fool tests into thinking they're prime, even though they're not. These pseudoprimes, known as Carmichael numbers, have been particularly difficult to grasp. It was only in the mid-1990s, for instance, that mathematicians proved there are infinitely many of them. Being able to say something more about how they're distributed along the number line has posed an even greater challenge.
Then along came Larsen with a new proof about just that, one inspired by recent epochal work in a different area of number theory. At the time, he was just 17.
[...] Mathematicians saw the potential for a perfect test of whether a given number is prime or composite. They knew that if N is prime, bN - b is always a multiple of N. What if the reverse was also true? That is, if bN - b is a multiple of N for all values of b, must N be prime?
Alas, it turned out that in very rare cases, N can satisfy this condition and still be composite. The smallest such number is 561: For any integer b, b561 - b is always a multiple of 561, even though 561 is not prime. Numbers like these were named after the mathematician Robert Carmichael, who is often credited with publishing the first example in 1910 (though the Czech mathematician Vaclav imerka independently discovered examples in 1885).
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