# Unusual behavior of 210 and 199 regarding prime numbers

Adding 210 to 199 over and over again, you get 8 more primes that can be arranged together into a 3x3 magic square:

1669 199 1249

619 1039 1459

829 1879 409

Is there any other pairs of numbers such as (210, 199) in the example above?

I asked this question to myself out of sheer curiosity, but couldn't find the right approach to answer it.

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You simply need nine primes in arithmetic progression to be able to do this - the magic square condition is irrelevant since since any $9$ numbers in arithmetic progression can be put into a magic square. – Thomas Andrews Aug 9 '14 at 22:51
Yeah, thanks, the magic number requirement is not crucial, its the consequence... – VividD Aug 9 '14 at 22:55

Any arithmetic progression of length $n^2$, $n \geq 3$ can be arranged into a magic square (because multiplying all the numbers in a magic square by a constant, or adding a different constant, doesn't affect the magicness of the square, and there are magic squares on $\{1, \dots, n^2\}$ for $n \geq 3$).
For example, the Wikipedia page I linked to notes that the numbers $468,395,662,504,823 + 205,619 · 223,092,870 · n$ are prime for $0 \leq n \leq 23$. So you could take the first nine of those numbers (or the second through tenth, and so on) and arrange them in a magic square of size $3$, just as you did with the arithmetic progression starting at $199$.
So we just need to break the current record by 2 to get a $5\times5$ magic square out of it. – Gerry Myerson Aug 10 '14 at 0:14