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Journal of Pure and Applied Mathematics

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Victor Sorokine*
Professor of Mathematics Mezos, France, Email: victor.sorokine2@gmail.com
*Correspondence: Victor Sorokine, Professor of Mathematics Mezos, France, Email: victor.sorokine2@gmail.com

Received: 12-Dec-2022, Manuscript No. puljpam-22-5923; Editor assigned: 14-Dec-2022, Pre QC No. puljpam-22-5923(PQ); Accepted Date: Dec 19, 2022; Reviewed: 15-Dec-2022 QC No. puljpam-22-5923(Q); Revised: 16-Dec-2022, Manuscript No. puljpam-22-5923(R); Published: 30-Dec-2022, DOI: 10.37532/2752-8081.22.6(6).11

Citation: Sorokine V. Fermat's last theorem: The numbers A, B, C are infinite. J Pure Appl Math. 2022; 6(6):11.

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In the base case of the Fermat equality, the k-digit endings of the numbers in the pairs (A^n, A), (B^n, B), (C^n, C) are equal to the k-digit endings of the numbers a^[n^(k-1)], b^[n^(k-1)], c^[n^(k-1)], where a, b, c are the last digits of the base numbers A, B, C and k is arbitrarily large, that is, the numbers A, B, C are infinite.


Fermat’s; Learning; Infinite



The a, b, c, p, q, r - the last digits in the numbers A, B, C, P, Q, R in the number system with a prime base n>2.

Ak-k-th (from the end) digit of the number A; A[k]-k-digit ending of number A.

Theorem (basic case of the FLT)

For prime n>2, and coprimes natural numbers A, B, C not multiple of n from the equality

1. An +Bn -Cn =0 and k>0

2. An [k]=A[k]=an^(k-1) [k], Bn [k]=B[k]=bn^(k-1) [k], Cn [k]=C[k]=cn^(k-1) [k].

The simplest properties of the equality 1*. Lemmas:

1. (dn-1)1=1 (where single digit d is not equal to 0). /Fermat's Little Theorem./

2. Numbers in pairs (C-B, P), (C-A, Q), (A+B, R) from equalities

3. [An =] Cn -Bn =(C-B)P, [Bn =] Cn -An =(C-A)Q, [Cn =] An +Bn =(A+B)R} are coprime and P[2]=Q[2]=R[2]=01 (because the digits p=q=r=1).

4. The digit (gn )t does not depend on gt. /Consequence from Binom Newton./

5. A[2]=an [2], B[2]=bn[2], C[2]=cn[2]. Corollary from 02°,

6. If A[k] = an^(k-1) [k], then An [k+1] = an^k [k+1]. Corollary from the Newton binomial for the number A=A°nk +an^(k-1) [k].

Proof of Theorem

Indeed, for k=1 the equalities 2* are a statement of a Little Theorem. For k=2, the equalities 2* follow from 03°, where P[2]=Q[2]=R[2]=01.

Subsequent digits with k>2 are calculated from the INFINITE series of Newton binomials for the numbers A, B, C, written as A=A°nk +an^(k-1) [k] (see 06°) with increasing k by 1 in each successive iteration.

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