Present prime number as primes!
Number Theory
Level
2
How many ways can 23 be represented as the sum or difference of two prime numbers?
2
23
46
32
0
1
8
11
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1 solution
Let us see the case for addition of 2 primes to give 23,and consider the same for subtraction.
Let there be 2 primes p 1 and p 2 satisfying the condition for addition(or subtraction). That is, p 1 + p 2 = 2 3 It is clear,that none of the primes is 2,as then it forces the other to be a composite number,21.
So, p 1 and p 2 both should be odd, at least.
But,WLOG, p 1 = 2 3 − p 2 which can be said equivalently informative for here as: p 1 = o d d − o d d = E V E N
But, p 1 [WLOG] can't be even,otherwise we know the consequences for the other prime as not being prime.
So here we get a conflicting situation:we need none of the 2 primes to be 2,but we need one of the 2 primes to be even. Therefore,the answer is : there are 0 such possible pairs(ways).