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Two Primes Make One Square

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

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:

121 and 289 not possible, all others written as sum of two prime numbers

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.