Friday, 7 August 2015

proof verification - Show that sqrtq is irrational


Let q be a positive integer such that q2 and such that for any

integers a and b, if q|ab, then q|a or q|b. Show that q
is irrational.




Proof;



Let assume q is a rational number, where n0 and gcd(m,n)=1, meaning q=mnq=m2n2



Since n2m2, q|m2q|m, so m=qt where tZ




By substitute m=qt in the equation qn2=m2, we get n2=qt2.



Since tells us that q|n2 and t2|n2, it contradicts with the assumption gcd(m,n)=1; therefore, q is irrational.



I get this proof with the assistant of the course, but is there any flaw or mistake? What are the other methods for proving this statement, can you at least give one different method? And how can I improve this proof?

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