Proof of Fermat’s Last
Theorem from Beal’s relation
Abstract: Multi-power relations with reference to Beal are
discussed with numerical examples. Some
additional properties of Beal’s conjecture are also identified to prove FLT
Keywords:
Fermat Last Theorem, Beal Conjecture, Number theory
Introduction
Fermat’s last
theorem says that integral solutions for ax +
by = cz are not possible when all the exponents are
same and greater than or equal to 3 . However it is possible if atleast one of
the exponents is different. In such multi-power relations, according to Beal
[1-3] the members when all the three members
a,b,c and the exponents x,y,z are
all positive integers such that (x,y,z) > 2, invariably have one or more
common factors. Conversely the equation
ax + by = cz has no solutions for any positive
integers a,b,c,x,y and z with (x,y,z)
> 2 for any set of co-prime
(a,b,c).The three members like or unlike power relations described by the Fermat’s assertion and Beal conjecture
respectively have some similarities and
differences.
The observed
similarities are (i) All the members can
be even, then the relation will be reducible. All the members cannot be odd due
to non-conservation of oddness (ii) One or two members but not all the three
may be square or higher power but all the members cannot be square or with the
same higher power, (iii) all the three members cannot be simultaneously primes
(iv) FLT and Beal conjecture are mathematically true with fractional and
rational or complex numbers, and (v) In irreducible form, both the FLT and Beal
conjecture do not have a common prime factor among the members
Among the
differences the most significant are (i) In ax + by = cz , FLT requires that a,b,c
cannot be positive integers when the exponents x,y,z are same and greater than
2, but they can be in Beal’s expression
when the exponents are different (ii) In FLt, a + b will always be
greater than c. but it can be greater, lesser or even be equal depending upon
the value of the exponents
In this
paper Beal’s expression in the form of an + an = 2m is studied to investigate more
properties of Beal’s conjecture and by using such properties FLT is
proved.
Multi-Power Relation of type an + an
= 2m
We know
that an + an = 2 an,
To make 2an to be a power of
a number the only possibility is “a” itself must be equal to 2x . Under this condition
it gives 2nx + 2nx
= 2nx+1 , where n and x have all positive whole integral values. For
different values of x we have
2n + 2n = 2n
+ 1
4n +
4n = 22n + 1
8n + 8n = 23n
+ 1
16n + 16n = 24n + 1
32n + 32n
= 25n + 1
All of
them have common prime factor 2 with which they can be reduced to an
irreducible form 1 + 1 = 2, which is a fundamental requirement for this type of
relations. Any power of 2 can be a common multiplier to generate multi-power
relations with bigger numbers. For example
2n + 2n
= 2n + 1 gives 22n+1 + 22n+1 = 4n+1 when all the members are
multiplied by 2n+1 .
This
method of getting multi-power relations, gives an ingenuous explanation for Fermat’s
Last Theorem. Sum of two identical cubes can be expressed as a square or fourth
or fifth power of a number, but never as a single cube or with any exponents of
multiples of three.
(2x2)3
+ (2x2)3 =(22x3)2 (2x4)3 + (2x4)3 = (2x3)4 (23x5)3 +
(23 x5)3 =
(22 x3)5
23 +
23 = 42 323 + 323 = 164 83
+ 83 = 45
83
+ 83 = 322 1623 + 1623 =
544 2563 + 2563 = 325
183 + 183 = 1082 5123 + 5123 = 1284 19443 +
19443 = 1085
When the
exponent of the sum is in the form of
3n+2, the Beal, expression of this type is
[22n+1
xn ]3 + [22n+1
xn ]3 = (22 x3)3n +2,
and it is (2n xn)3 + (2n xn)3
= (2x3) 3n+1 when the exponent is in the form of 3n + 1. It
is noted that the exponent of the sum will never be in the form of 3n.. To be
true, it should satisfy the relation (2n
xn)3 + (2n
xn)3 = (2m x)3n which necessitates the condition 23n + 1 = 23mn
No integer
values for n and m satisfy this condition, which substantiates the Fermat’s
last Theorem without any further argument.
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