ZAP
Note the numbers 294,001, 505,447 and 584,141. Do you notice anything special about them? You may recognize that they are all primes: divisible only by themselves and 1. But these particular primes are even more unusual.
Mathematicians have revealed a new category of prime numbers “digitally delicate”.These cousins, which exist in infinite numbers, cease to be faster than Cinderella at midnight when undergoing a change in any of its digits.
Os digitally delicate cousins have a very particular property: if you change any of their digits to another value, the result then becomes a composite number.
To use a simpler example, let’s consider 101which is a prime number. If we change the digits to get 201, 102 or 111, we now have values divisible by 3 and therefore composite numbers.
Although the idea is decades old, mathematicians at the University of South Carolina have identified an even more specific niche inside the digitally delicate cousins: the largely digitally delicate cousins.
It is about prime numbers to which they are addedconceptually, unlimited leading zeroswhich does not change the original value of the prime, but makes a difference when these zeros become other digits to test this delicacy, explains .
So instead of 101, let’s think about 000101. This value remains prime, and the zeros are there only as an additional representation. But if we change one of these zeros — for example, turning 000101 into 100101 — we get a composite number, divisible by 3.
According to mathematicians, there is infinite digitally delicate primesbut until now failed to present a single example concrete. They have already tested all prime numbers up to 1,000,000,000 by adding zeros to the left and doing the calculations.
The math teacher Michael Filasetafrom the University of South Carolina, and former graduate student Jeremiah Southwick worked together on an investigation into largely digitally sensitive numbers. His work was featured in a published in the journal Mathematics of Computation.
Even without specific examples, the two mathematicians managed to prove that these numbers exist in base 10 — that is, in the number system that uses the digits from 0 to 9, unlike the binary system, with base 2, which uses only 0 and 1 — and which exist in infinite quantity.
The proof is based on a type of logic that resembles, in a much more sophisticated version, the simple divisibility rules.
Certain families of numbers, such as those containing 9 or whose sum of digits reaches a certain value, can be demonstrated in a general way and then distributed among different “groups”. The more groups there are, the greater the part of the gigantic set of integers that is “covered” by the proof.
The situation involving the largely digitally delicate cousins is, of course, more complicated, explains . Many more groups are neededsomething like 1.025.000and in one of these groups any prime number is guaranteed to become composite if any of its digits, including the leading zeros, are increased.
This is not the kind of mathematics that leads to immediate practical application: it is number theory that is valid mainly for itselfas a way of exploring the limits of mathematics.
And since Filaseta and Southwick published their demonstrations, they have already begun to more special cases of digitally sensitive numbers emergeas other mathematicians use this work as a starting point.
What if we took 101 and plugged a 1 into it, getting 1011? What if we removed one digit, leaving us with 10? The possibilities are digitally limitless.
