Entegers

Entegers

Postby Larry » Wed Jun 09, 2021 9:11 am

Do not know if these mathematical objects have been explored before but their properties are interesting to me.

They are rational numbers of the form (1 + 1/N) where N is a nonzero positive integer. If they are represented by N and treated as entities unto themselves then we can easily prove that the product of a sequence of these "Entegers" from N = 1 to K is K+1:
1 x2x...K = 1 x 2 x ... K = K+1.

A shorthand for such a sequence can be [N] where it is understood that N starts at 1 goes to N so [N] = N+1

I call these things Entegers because [tex]\lim_{N\to \infty} N^{N}= e[/tex] (the base of natural logarithms)

It was interesting to find that any rational number > 1 can be represented by a unique sequence of these things.

Let a, b be positive rational (nonzero) and say b < a so the R = a/b. Since [a-1] = a and [[b-1]b[/b]] = so
[[b]a-1
]/[b-1] = [b, a-1] would be the consecutive sequence.

Have such things been explored before?
Larry
 
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