The Irrational Number Generator

It is stated (although the proof is not clearly demonstrated), that "if an integer is not an exact k power of another integer then its k root is irrational" . What follows below is a proof that shows that this is generally true even for the more difficult case of fractions, a fact that was not apparent until a proof for Fermat’s Last Theorem was recently found.

Theorem

For natural numbers n and integers a, b, the nth Root of

[(b/a)^n + 1]

is irrational for n > 2. . Hence this formula can be used to generate an infinite number of irrational numbers.

Proof

Assume that the nth Root of [

(b/a)^n + 1]

is rational, then so is nth Root

[(b/a)^n + 1]*a

Hence, nth Root [

(b/a)^n + 1

] = c/q . . . . . . for some integers c and q

So,

a^n + b^n = (c/q)^n

And

q^n * a^n + q^n * b^n = c^n

. . . . . let d = q * a and e = q * b,

Thus,

d^n + e^n = c^n

, which since d and e are integers, contradicts “Fermat’s Last Theorem” which has recently been proved by Andrew Wiles. Hence nth Root

[(b/a)^n + 1]

must be irrational, for n > 2.

NB this result was already known for the case where b/a is actually a whole number (due to the fundamental theorem of arithmetic and the fact that the nth root of primes are irrational), and in this respect provides an alternative proof.. However this was not previously known to be true for fractions, as demonstrated above. For example if we take 16, which is a square of 4 and add 1, the square root is irrational. However if we take a 4 and b 3, the fraction 3/4 when squared and added to one, does not yield an irrational number when square rooted. This result can only occur when n = 2 but the process will always produce an irrational number for n > 2. Indeed, if the above theorem could be shown using an alternative method, it would supply a rather quick proof of Fermat’s Last Theorem.

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