![]() But that didn’t prevent them from proving some strong statements about these hypothetical numbers.įirst, they showed that such numbers are indeed possible in base 10, and, what’s more, an infinite number of them exist. In fact, Filaseta and Southwick couldn’t find one example in base 10 of a widely digitally delicate prime, despite looking through all the integers up to 1,000,000,000. “294,001 is digitally delicate, but not widely digitally delicate,” Pollack said, “since if we change …000,294,001 to …010,294,001, we get 10,294,001” - another prime number. ![]() Not surprisingly, the added condition makes such numbers harder to find. The numbers 53 and …0000000053 have the same value, after all would changing any one of those infinite zeros tacked on to a digitally delicate prime automatically make it composite?įilaseta decided to call such numbers, assuming they existed, “widely digitally delicate,” and he investigated their properties in a November 2020 paper with his former graduate student Jeremiah Southwick. Motivated by Erdős’ and Tao’s work, Filaseta wondered what would happen if you included an infinite string of leading zeros as part of the prime number. “It’s a remarkable result,” said Paul Pollack of the University of Georgia. Now, with two recent papers, Michael Filaseta of the University of South Carolina has carried the idea further, coming up with an even more rarefied class of digitally delicate prime numbers. That means the average distance between consecutive digitally delicate primes remains fairly steady as prime numbers themselves get really big - in other words, digitally delicate primes won’t become increasingly scarce among the primes. Other mathematicians have since extended Erdős’ result, including the Fields Medal winner Terence Tao, who proved in a 2011 paper that a “positive proportion” of primes are digitally delicate (again, for all bases). He proved not only that they do exist, but also that there are an infinite number of them - a result that holds not just for base 10, but for any number system. His question got a quick response from one of the most prolific problem solvers of all time, Paul Erdős. ![]() ![]() In 1978, the mathematician and prolific problem poser Murray Klamkin wondered if any numbers like this existed. Such numbers are called “digitally delicate primes,” and they’re a relatively recent mathematical invention. Change the 1 in 294,001 to a 7, for instance, and the resulting number is divisible by 7 change it to a 9, and it’s divisible by 3. If you pick any single digit in any of those numbers and change it, the new number is composite, and hence no longer prime. Notice anything special about them? You may recognize that they’re all prime - evenly divisible only by themselves and 1 - but these particular primes are even more unusual. ![]()
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