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induction proof of fundamental theorem of arithmetic

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Background: In 1988 I read the book ”one, two, three, infinity ” by George Gammow. The book had a statement to the effect that no polynomial had been found such that it generates all the prime numbers and nothing but prime numbers. This was true at the time Gammow wrote the book; however subsequently a polynomial was constructed fulfiling the condition given above. I then experimented with some polynomials and found that although one cannot generally predict the prime numbers generated by a polynomial one can predict the composite numbers generated by a polynomial. Since I was originally trying to predict the primes generated by a given polynomial (which may be called ”successes ”) but could predict the ”failures” (composite numbers) I called functions which generate failures ”failure functions ”. I presented this concept at the Ramanujan Mathematical society in May 1988. Subsequently I used this tool in proving a theorem similar to the Ramanujan Nagell theorem at the AMS-BENELUX meeting in 1996.

Abstract definition: Let f(x)fxf(x) be a function of xxx. Then x=g(x0)xgsubscriptx0x=g(x_{0}) is a failure function if f(g(x_0)) is a failure in accordance with our definition of a failure.Note: x0subscriptx0x_{0} is a specific value of xxx.

Examples: 1) Let our definition of a faiure be a composite number. Let f(x)beapolynomialinxwherexbelongstofxbeapolynomialinxwherexbelongstof(x)beapolynomialinxwherexbelongsto Z.Thenfragmentsnormal-.Then.Thenx==x_0 + kf(x_0) is a failure function since these values of xxx are such that f(x) are composite.

2) Let our definition of a failure again be a composite number. Let the function be an exponential function ax+cwhereaandxbelongtoN,cbelongstoZsuperscriptaxcwhereaandxbelongtoNcbelongstoZa^{x}+cwhereaandxbelongtoN,cbelongstoZ and a and c are fixed. Then x=x0+k*Eulerphi(f(x0)fragmentsxsubscriptx0kEulerphifragmentsnormal-(ffragmentsnormal-(subscriptx0normal-)x=x_{0}+k*Eulerphi(f(x_{0}) is a failure function.Here also x0subscriptx0x_{0} is fixed. Here k belongs to N.

3) Let our definition of a failure be a non-Carmichael number. Let the mother function be 2n+49superscript2n492^{n}+49. Then n=5+6*kn56kn=5+6*k is a failure function. Here also kkk belngs to NNN.

Applications: failure functions can be used for a)fragmentsanormal-)a) indirect primality testing and b)fragmentsbnormal-)b) as a mathematical tool in proving theorems in number theory.

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