## High Sensitivity Primes

Even if they haven't been able to find a concrete example of this, mathematicians have proved the existence of a widespread type of prime number. However, their indivisibility is lost with the slightest change.

294 001, 505 447 and 584 141: Notice something special about these numbers? You may have realized that numbers are prime, but they actually have another amazing property. If you change one of the digits to any other, then suddenly (besides one and itself) they have additional divisors. For example, if you replace the 1 in 294,001 with a 7, the result can be divided completely by seven; If you turn 1 into 9, the result is divisible by three.

Numbers like this are called weak prime numbers – and they are relatively young research objects. In 1978, the mathematician Murray Klamkin (1921-2004) first asked himself whether prime numbers with such characteristics existed. His famous Hungarian colleague Paul Erdős (1913-1996) found an answer shortly afterwards. He proved that there are infinitely many weak prime numbers - in any number system, for example among the base-two binary numbers. Since then, there have been several advances in the field. For example, as Fields Medalist Terence Tao showed in 2011, a "positive fraction" of primes is weak, meaning their average spacing stays about the same-they don't get rarer as primes grow.

In light of these results, Michael Filaseta of the University of South Carolina has further developed the concept and has encountered a new class of prime numbers. He wondered what happens when you include an infinite string of leading zeros, so instead of 53, consider the number …0000000053. Does it necessarily gain in divisors as soon as you replace any of the zeros or the other digits with any different value?

Spektrum der Wissenschaft's translated and edited version of the article "Mathematicians Find a New Class of Digitally Delicate Primes" from Quanta Magazine, a content independent magazine of the Simons Foundation dedicated to the dissemination of research in mathematics and the natural sciences.