# Thread: n = p + a^2

1. ## n = p + a^2

I want to disprove the following:

"Every positive integer is expressible in the form p+a^2 where a is an integer, and p is either prime or 1."

(Note: I had a nice, easy counterexample until I noticed that, defying all my prior experience, the text includes all "negative primes" as being prime--not just natural number primes).

2. Originally Posted by mylestone
I want to disprove the following:

"Every positive integer is expressible in the form p+a^2 where a is an integer, and p is either prime or 1."

(Note: I had a nice, easy counterexample until I noticed that, defying all my prior experience, the text includes all "negative primes" as being prime--not just natural number primes).

one example is $13^2=169.$ here's how to construct more examples:
choose any integer $b > 1$ such that $2b \pm 1$ is not prime (for example $b=18c^2-6c+1, \ c \geq 1$). suppose $b^2=p+a^2,$ for some integers $a$ and $p$ with $p$ either 1 or prime (positive or negative).
then $(b-a)(b+a)=p.$ so either $b-a=\pm 1, \ b +a=\pm p,$ or $b-a=\pm p, \ b+a= \pm 1.$ thus: $2b \mp 1 = \pm p,$ which is impossible because $b>1$ and $2b \pm 1$ was assumed to be non-prime. Q.e.D.