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Great Proofs

√2 is irrational

This proof, a standard bearer for all reductio ad absurdum proofs, is one of the early mathematics proofs - said to be from around 500 B.C. from the followers of Pythagoras.

A 'rational' number is a fraction a/b, where a and b are integers.

Another elegant proof:
In 1952, Robert Gauntt, then a freshman at Purdue University, gave the following proof[2].
a2 = 2b2 cannot have a non-zero solution in integers because the last non-zero digit of a square, written in base 3, must be 1, whereas the last non-zero digit of twice a square is 2.
(See multiplication in any base!)

See HERE for other proofs and generalizations.

Remark
Robert Gauntt used this result: The last non-zero digit of a square, written in base 3, must be 1.
A more general statement is the following:
The last digit of the square of any positive integer written in base 3 can not be 2.
A proof of this is immediate on noting that any positive integer n can be written as n = 3q + r, where q and r are integers and r is one of the integers 0,1, or 2. Then n2 = 9q2 + 6rq + r2. If r is 0 or 1, then r2 = r = 0 or 1. If r = 2, then n2 = 3q'+ 1. This means, in base 3, n2 ends in a 0 or 1.

Similarly one can prove the following.
The last digit of the square of any positive integer in base 5 can not be 2 or 3.
The last digit of the square of any positive integer in base 10 can not be 2,3,7,or 8.

Gauntt used the result for base 3 noting further that the last non-zero digit of the square must be 1.

REFERENCES
[1]G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Oxford University Press, 1978
[2] American Mathematical Monthly, Vol 63 (1956), P. 247

(G.R.T.)

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