An algebra problem by Alf-archie Sangkula
Algebra
Level
pending
Find the sum of all primes that can be written both as a sum of two primes and as a difference of two primes.
The answer is 5.
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1 solution
Let the prime in question (i.e. the one that is both a sum of two primes and a difference of two primes) be denoted by p.
Since the problem talked about sums and differences of primes, the first thing that came to mind was that all primes except 2 are odd. If two odd primes are added together, then their sum is even, and since the sum is clearly bigger than 2, it cannot be prime. This implies that p must be of the form q + 2, where q is an odd prime.
Similarly, if one odd prime is subtracted from another, the difference must be even. Hence, in order for the difference to be a prime, it must be 2. However, p cannot be 2 since p is also the sum of two primes. This implies that p must be of the form r - 2, where r is an odd prime.
Putting this all together, we find that p-2, p and p+2 must all be prime numbers. However, anyone who’s listed out primes before will know that 3 primes cannot occur so closely to each other like that: one of them will definitely be divisible by 3*. Hence, the only way all 3 of them could possibly be prime is if the number divisible by 3 is 3 itself, i.e. p = 5.
Since 5 = 3 + 2 and 5 = 7 – 2, 5 indeed is the only prime that can be written both as the sum of two primes and as a difference of two primes.