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We argue that prime numbers have no rational square roots.

For integers p, and q  0, the ratio of p to q is the square root of an integer a just when pp = aqq.

We shall consider only the non-negative integers here, so we must show that

Thm*  a:. prime(a (p:q:pp = aqq)

It is impossible to build an infinite decreasing sequence of natural numbers, as is expressed by our principle

Thm*  P:(Prop). (x:P(x (x':x'<x & P(x')))  (x:P(x))

So, to show that a prime number a   has no rational square root, it is enough to show that if you could find a ratio "p/q", expressed with a non-negative numerator, whose square was a, then you could find a smaller such numerator p' of a rational root "p'/q'" i.e.,

p:q:pp = aqq  (p':p'<p & (q':p'p' = aq'q'))

(see hypothesis Hyp:6 of the proof)

We rewrite

(pp = aqq) to (qq = ap'p')

and then

(qq = ap'p') to (p'p' = aq'q')

giving us a rational square root with numerator p'<p and denominator q'. EXAMPLE

These rewrites are justified by a special-purpose lemma

Thm*  a:
Thm*  prime(a (p:q:pp = aqq  (p':p'<p & qq = ap'p'))

See the GLOSS of the lemma.

QED

There is also a specialized proof for a = 2. It is simpler but similar.
Thm*  (p:q:pp = 2qq) with a gloss.

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PfPrintForm Definitions PrimesSquareRoots Sections DiscrMathExt Search Doc