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Thread: Fermat's Theroem and the Relativistic Unit Circle

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    Default Fermat's Theroem and the Relativistic Unit Circle

    The Fermat Circle

    See also

    The Relativistic Unit Circle

    The Binomial Expansion yields the result:

    where rem(a,b,n)> 0.

    This by itself proves Fermat's Theorem for a,b, and c integers and n >2

    (for n = 2, the expression can be that of a circle if a,b,c form a Pythagorean triple; otherwise the Binomial Theorem applies.

    If Fermat's expression , it would mean that rem(a,b,n)= 0.

    Since Rem(a,b,n) consists of additive terms which are all composed of multiplicative products of a and b
    (e.g. , it can only vanish if a=0 or b=0, so that

    or , respectively

    Since the Binomial Theorem is valid for all positive numbers (e.g. fractions, transcendentals) and even complex numbers, the question of whether a and b are integers is moot - it is the remainder term that must vanish.

    ---------------------------------------

    For the relativistic circle in two dimensions (a,b) positive numbers are generated in the positive quadrant of the circle by the prescription when , and vice versa, so that n = or at or at v=c, respectively.

    (for v=0, , , but at , , , so for each integer generated represented by a, and .

    (for v = c, , , and b=b for each integer b.

    The product can bet thought of as the area of either a concentric circle or the diagonal of an inscribed polygon, where the area of its equivalent square is

    For , the product cannot be an integer, since , corresponding to the result of the Binomial Expansion of , since

    For the corresponding expression is:

    which can only be valid for and (i.e.,

    (For QFT, it is instructive to think of or = h, where h is Planck's constant).
    Last edited by BuleriaChk; 12-26-2016 at 04:12 PM.
    _______________________________________
    "Flamenco Chuck" Keyser
    The Relativistic Unit Circle 03/28/2017 07:40 AM PST
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    Default Re: Fermat's Theroem and the Relativistic Unit Circle

    Thread Merging.
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