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

**Victor Sorokine**

^{*}**Email:**victor.sorokine2@gmail.com

**Victor Sorokine, Professor of Mathematics Mezos, France,**

^{*}Correspondence:**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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## 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. A^{n} +B^{n} -C^{n} =0 and k>0

2. A^{n} _{[k]}=A_{[k]}=a^{n^(k-1)} _{[k]}, B^{n} _{[k]}=B_{[k]}=b^{n^(k-1)} _{[k]}, C^{n} _{[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. [A^{n} =] C^{n} -B^{n }=(C-B)P, [B^{n }=] C^{n} -A^{n} =(C-A)Q, [C^{n }=] A^{n }+B^{n} =(A+B)R} are coprime and P[2]=Q[2]=R[2]=01 (because the digits p=q=r=1).

4. The digit (g^{n} )_{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 A^{n} _{[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.