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## 'Two Primes Make One Square' printed from http://nrich.maths.org/

This question has been very well answered - thank you to
everyone who sent us solutions. David, Jack and Matthew, and Beth,
Harry and Henry from St. Nicolas CE Junior School, Newbury explain
how they tackled it:

We listed the prime numbers up to 100 and then we
listed the squares of the numbers between 4 and 20. Then we started
working out the answers by adding two prime numbers together to
make the squares until we reached 400. We found out that two
squares were impossible to make. When we had an odd square number,
we had to have a number that adds on to 2 to make the square number
and sometimes this number wouldn't be prime. For example, to make
121 with two primes, one of the primes has to be 2, and 119 is not
prime.

Our results were:

25 = 2 + 23

36 = 13 + 23

49 = 2 + 47

64 = 17 + 47

81 = 2 + 79

100 = 3 + 97

121 impossible

144 = 47 + 97

169 = 2 + 167

196 = 29 + 167

225 = 2 + 223

256 = 89 + 167

289 impossible

324 = 101 + 223

361 = 2 + 359

400 = 41 + 359

Martha from Tattingstone School went about it in a slightly
different way, but came to the same conclusions:

Firstly I looked to see what the square of 20 was, 400. Then I
found the nearest prime number that was below 400 and found it was
397. I then took 397 away from 400 and found it was 3 which is also
a prime number so I added them together to get 20². I
did that on all of them and got:

I noticed that all the odd numbers that needed to be squared
e.g. 19, 17, 15, 13 etc. needed to be added by two because it is
the only even number that is a prime number as well.

Thanks to Alice and James from Tattingstone, also William from
Sydney Grammar School, St Ives who sent in correct answers as
well.