# 4x4 magic square consisting of consecutive composite numbers

Is it possible to create a fourth-order magic square consisting of consecutive composite numbers that don't form an arithmetic sequence? If possible, give an example . If not, provide a proof.

Clarification:

In case someone is not sure what consecutive composite numbers are, here is an example: 4, 6, 8, 9 and 10 are five consecutive composite numbers because they are all the composite numbers from 4 to 10 and they are listed in order.

• Doesn't every set of consecutive numbers form an arithmetic sequence? Or do you mean that for example 4 and 6 are consecutive because we're only looking at composite numbers? – Jaap Scherphuis May 13 at 8:48
• @JaapScherphuis See edit. – Peđa Terzić May 13 at 8:59

It is possible.

8  11 14 1
13 2  7  12
3  16 9  6
10 5  4  15

You can then transform it by

decrementing the numbers 1 to 8

which gives the following magic square which is not in arithmetic progression:

7  11 14 0
13 1  6  12
2  16 9  5
10 4  3  15

Now we just need to add the right number to make them all composite:

It has numbers 0-7 and 9-16, missing number 8. We need to find a prime p with a prime gap of at least 9 on both sides, and then add p-8. This makes the missing number the prime p, and all the other numbers composite. The first such prime is 211.

210   214   217   203
216   204   209   215
205   219   212   208
213   207   206   218

This has magic constant

844