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Thread: n = p + a^2

  1. #1
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    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).

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by mylestone View Post
    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).

    Thanks in advance.
    one example is $\displaystyle 13^2=169.$ here's how to construct more examples:

    choose any integer $\displaystyle b > 1$ such that $\displaystyle 2b \pm 1$ is not prime (for example $\displaystyle b=18c^2-6c+1, \ c \geq 1$). suppose $\displaystyle b^2=p+a^2,$ for some integers $\displaystyle a$ and $\displaystyle p$ with $\displaystyle p$ either 1 or prime (positive or negative).

    then $\displaystyle (b-a)(b+a)=p.$ so either $\displaystyle b-a=\pm 1, \ b +a=\pm p,$ or $\displaystyle b-a=\pm p, \ b+a= \pm 1.$ thus: $\displaystyle 2b \mp 1 = \pm p,$ which is impossible because $\displaystyle b>1$ and $\displaystyle 2b \pm 1$ was assumed to be non-prime. Q.e.D.
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