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Sheldon conjecture solved

The 73rd episode of the US sitcom "The Big Bang Theory" has long been a special one for mathematicians: "What is the best number that is known?" Asks Sheldon Cooper at one point. "It's 73," says the ingenious but unsuitable physicist himself in the end.

Sheldon's reasoning is a feast for fans of numbers: "73 is the 21st prime number, its mirror number - 37 - the 12th, and its mirror number (21) is the product of the multiplication of - hold on tight: 7 and 3." What caused laughter among the other series characters and many viewers, made professional mathematicians ponder: Are there more so-called Sheldon primes that share exactly such properties?

This article is featured in Spectrum Compact, Prime Numbers - The Stars of Mathematics

The number theorist Carl Pomerance from Dartmouth College in New Hampshire, together with his colleague Christopher Spicer from Morningside College in Iowa, has now found an answer: The 73 is actually the only prime number that meets the criteria mentioned by Sheldon, the researchers write in an im » American Mathematical Monthly ”publication.

Shortly after the broadcast of "The Big Bang Theory" episode in 2015, Spicer, together with two colleagues, defined a Sheldon prime as that n. Prime number pnfor which the product of their digitsn and its mirror number rev (pn) the rev (n). Prime numberprev (n) results. To put it a bit more understandably, that means for them xyz. Prime number abcd: On the one hand results a * b * c * d = xyz, and besides is dcba the zyx. Prime number. When the three researchers tested the first ten million prime numbers for these properties, they found that only the 73 satisfies both at the same time. They then suggested that there was just this one Sheldon prime number.

The final proof from Pomerance and Spicer took years to come. In a first step, the two mathematicians showed that there can be no Sheldon prime greater than 1045 is. They come to this conclusion thanks to the famous prime number theorem from 1896, which specifies the minimum number of all prime numbers in a certain number interval. Indeed, the condition can be that the product of all digits of a Sheldon prime numberpn the numbern results for numbers greater than 1045 can no longer be fulfilled. Because in such a case the number isn of the prime numbers in the interval [2,pn] is always greater than the product of the digits of according to the prime number theorempn.

This assessment is the central point of the work. Because even if 1045 is an inconceivably large number, it is nevertheless finite. This means that you can systematically find all prime numbers between 2 and 1045 hack it with a computer to look for more Sheldon primes. But of course you can't do it without tricks. Having an algorithm examine numbers with 45 digits is a challenge even for the best hardware. Therefore, Pomerance and Spicer restricted the possible Sheldon candidates with the help of the required properties, and also used approximation formulas to approximate extremely large prime numbers with integrals , and gradually sorted out all Sheldon candidates - until at some point only the 73 remained.