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Math Help - 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 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.
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