## The arithmetic sequence, $1487, 4817, 8147$, in which each of the terms increases by $3330$, is unusual in two ways: (i) each of the three terms are prime, and, (ii) each of the $4$-digit numbers are permutations of one another.

There are no arithmetic sequences made up of three $1$-, $2$-, or $3$-digit primes, exhibiting this property, but there is one other $4$-digit increasing sequence.

What $12$-digit number do you form by concatenating the three terms in this sequence?

### To find the other 4-digit increasing arithmetic sequence, we need to find three 4-digit prime numbers which are permutations of each other and have a common difference.

We can use computer algorithms to perform the brute-force checking, going through all 4-digit prime numbers and checking for these two conditions. Once we identify the three numbers, we can concatenate them to form a 12-digit number.

The three primes that fit this requirement are $2969$, $6299$, and $9629$. These three prime numbers have a common difference of $3330$ and are permutations of each other.

By concatenating these three terms, we obtain the 12-digit number we are looking for: $296962999629$.

##### More Answers:

Goldbach’s Other ConjectureDistinct Primes Factors

Self Powers

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