Fermat's Last Theorem: the Numbers A, B, C Are Infinite

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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Abstract

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.

Keywords

Fermat’s; Learning; Infinite

Introduction

Designations

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.

A_k-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]=a^{n^(k-1)} _[k], Bⁿ _[k]=B_[k]=b^{n^(k-1)} _[k], Cⁿ _[k]=C_[k]=c^{n^(k-1)} _[k].

The simplest properties of the equality 1*. Lemmas:

1. (d^n-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] = a^{n^(k-1)} _[k], then Aⁿ _[k+1] = a^{n^k} _[k+1]. Corollary from the Newton binomial for the number A=A°n^k +a^{n^(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°n^k +a^{n^(k-1)} _[k] (see 06°) with increasing k by 1 in each successive iteration.