What is a squime?

Question

What is a squime?

Answer 1

A squime is a number with the following properties:

- Digits are used no more than once

- Each two adjacent digits are either a square or a prime number.

The term was used in a question of a contest of the Australian Mathematics Trust in 2005. The question was to find the largest squime.

Answer: 25973641

The reasoning goes something like this.

The largest squime has at most eight digits, since it cannot contain 8 or 0. If 2 is to be used, it must come first because no square or 2-digit prime ends in 2. No 2-digit prime ends in 5 and only 25 is square. If, therefore, we use 5, it must come after 2. So we begin with 25.

No 2-digit square begins with 5, but 59 is prime, so 9 is the largest digit that can follow 25. Now we have 259.

No 2-digit square begins with 9, but 97 is prime, so 7 is the largest digit to follow 259. Now we have 2597.

Continuing in this way, we find that 25973641 is the largest squime.