In 1988, Florida International University mathematician T.I. Ramsamujh offered a proof that all positive integers are equal. “The proof is of course fallacious but the error is so nicely hidden that the task of locating it becomes an interesting exercise.”

Let

p(n) be the proposition, ‘If the maximum of two positive integers isnthen the integers are equal.’ We will first show thatp(n) is true for each positive integer. Observe thatp(1) is true, because if the maximum of two positive integers is 1 then both integers must be 1, and so they are equal. Now assume thatp(n) is true and letuandvbe positive integers with maximumn+ 1. Then the maximum ofu– 1 andv– 1 is n. Sincep(n) is true it follows thatu– 1 =v– 1. Thusu=vand sop(n+ 1) is true. Hencep(n) impliesp(n+ 1) for each positive integern. By the principle of mathematical induction it now follows thatp(n) is true for each positive integern.Now let

xandybe any two positive integers. Takento be the maximum ofxandy. Sincep(n) is true it follows thatx=y.

“We have thus shown that any two positive integers are equal. Where is the error?”

(T.I. Ramsamujh, “72.14 A Paradox: (1) All Positive Integers Are Equal,” *Mathematical Gazette* 72:460 [June 1988], 113.)