# Crossnumber winners, Issue 08

Did you solve it?

It’s that time of the year again: time to announce the winners of the Chalkdust prize crossnumber #8! Before we reveal the winners, we can reveal the solution and some interesting bits of maths hidden in the clues.

### The solution

The solution of crossnumber #8 is:

 1 7 1 4 2 1 5 7 4 3 3 7 1 3 3 7 8 2 8 5 3 3 0 3 6 7 7 1 1 7 7 7 1 7 7 7 5 8 3 2 7 2 4 2 3 6 7 4 0 2 5 7 1 9 3 1 5 3 4 7 6 4 0 1 4 4 4 3 4 6 4 3 3 6 7 1 1 1 8 6 5 7 6 0 7 1 7 6 0 7 8 5 5 7 1 1 7 1 1 3 7 7 7 5

The sum of the across clues was 43,788,970.

### Some maths

As usual, there were some nice bits of maths hidden in the clues.

Clues 2D, 25D, and 19A were all palindromes. 12A was the sum of these three palindromes. You might have thought that this would limit the number of options for 12A, but you can actually write any integer is the sum of three palindromes. If you want to try this out yourself, Christian Lawson-Perfect has made a webpage that can tell you how to write any number as the sum of three palindromes.

Clues 6D, 5D, 23A, 26D, 8D, 22D, 4D, 3D, 20D, 36A, and 9D formed a sequence of prime numbers where each prime was 16 more than three times the previous. This sequence is (obviously) on the OEIS.

Clues 1A, 31D, and 28D asked for a numbers $n$ such that the largest prime factor of $n^2-1$ is 3, 5 or 7. Surprisingly, there are only a finite number of choices for each of these: in fact, for any prime number $p$ there is a largest number $k$ such that $k^2-1$’s largest prime factor is $p$. There’s more information about this on the OEIS.

Clue 34A asked for a number $k$ such that $k\times n^2+1$ is never prime for any integer $n>0$. Such numbers are called Sierpiński numbers. It is conjectured (but not proven) that 78557 (the solution of this clue) is the smallest Sierpiński number. Again, there’s more info on the OEIS.

Clue 21A asked for a prime number of the form $n^n+1$. The only known prime numbers of this form are 2, 5, 257 ($n=1,2,4$), although it is not known whether or not there are more primes of this form. Obligatory OEIS link.

### Picking the winners

We received 101 entries, 89 of which were correct.

The winner of the Chalkdust Issue 08 crossnumber competition is Hannah Steiner, who wins a £100 goody bag from Maths Gear. The runners-up, and winners of Chalkdust T-shirts, are Hector Bouton, Lewis Dyer and Paul Appleby.

Thank you to everyone who entered the crossnumber competition! The competition will return in Issue 09 of Chalkdust, released in March…

Matthew is a postdoctoral researcher at University College London. He hasn’t had time to play Klax since the noughties, but he’s pretty sure that Coke is it!

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