scholarly journals I. Continuation of the subject of a paper read Dec. 22, 1853, the supplement to which was read Jan. 12, 1854, by Sir Frederick Pollock, &c.; with a proof of Fermat's first and second theorems of the polygonal numbers, viz. that every odd number is composed of four square numbers or less, and of three triangular numbers or less

1856 ◽  
Vol 7 ◽  
pp. 1-4
Keyword(s):  
Triangular Numbers ◽  
The Subject ◽  
Polygonal Numbers ◽  
Four Square

The object of this paper is in the first instance to prove the truth of a theorem stated in the supplement to a former paper, viz. “that every odd number can be divided into four squares (zero being considered an even square) the algebraic sum of whose roots (in some form or other) will equal 1, 3, 5, 7, &c. up to the greatest possible sum of the roots.” The paper also contains a proof, that if every odd number 2 n + 1 can be divided into four square numbers, the algebraic sum of whose roots is equal to 1, then any number n is composed of not exceeding three triangular numbers.

scholarly journals Polygonal Numbers

2013 ◽  
Vol 21 (2) ◽  
pp. 103-113 ◽  
Author(s):  
Adam Grabowski
Keyword(s):  
Related Result ◽  
Integer Sequences ◽  
Triangular Numbers ◽  
Perfect Numbers ◽  
Polygonal Numbers ◽  
Online Encyclopedia ◽  
Four Square ◽  
Mersenne Primes ◽  
Ordinal Index

Summary In the article the formal characterization of triangular numbers (famous from [15] and words “EYPHKA! num = Δ+Δ+Δ”) [17] is given. Our primary aim was to formalize one of the items (#42) from Wiedijk’s Top 100 Mathematical Theorems list [33], namely that the sequence of sums of reciprocals of triangular numbers converges to 2. This Mizar representation was written in 2007. As the Mizar language evolved and attributes with arguments were implemented, we decided to extend these lines and we characterized polygonal numbers. We formalized centered polygonal numbers, the connection between triangular and square numbers, and also some equalities involving Mersenne primes and perfect numbers. We gave also explicit formula to obtain from the polygonal number its ordinal index. Also selected congruences modulo 10 were enumerated. Our work basically covers the Wikipedia item for triangular numbers and the Online Encyclopedia of Integer Sequences (http://oeis.org/A000217). An interesting related result [16] could be the proof of Lagrange’s four-square theorem or Fermat’s polygonal number theorem [32].

III. On Fermat’s theorem of the polygonal numbers, with supplement

1864 ◽  
Vol 13 ◽  
pp. 542-545
Keyword(s):  
Triangular Numbers ◽  
Polygonal Numbers ◽  
Four Square

This paper (with its Supplement) proposes a proof of the first two theorems of Fermat, relating to the polygonal numbers, viz. that every number is composed of not exceeding three triangular numbers, and not exceeding four square numbers. And this is done by a method entirely new, founded on the properties of the triangular numbers and the square numbers, and the relation they bear to each other, and on the expansion of an algebraical expression of three members into a line , a square , and a cube , so as to obtain every possible value of the whole expression; and throughout the proof every number or term in a series (except in the Table) is expressed by the roots of the squares that compose it, and the roots only are dealt with, and not the numbers or the squares that compose them; a Table is constructed from the triangular numbers, thus (see opposite page). Mode of constructing the Table. The series of triangular numbers is in the centre of the Table. Below that series the adjoining terms are united, and they form the square numbers 1, 4, 9, &c.; the next adjoining terms are united, and they form the next row, and so on.

scholarly journals 1. On the extension of the principle of Fermat’s theorem of the polygonal numbers to the higher orders of series whose ultimate differences are constant. With a new theorem proposed, applicable to all the orders

1851 ◽  
Vol 5 ◽  
pp. 922-924 ◽  
Keyword(s):  
General Expression ◽  
Triangular Numbers ◽  
Polygonal Numbers

The object of this paper professes to be to ascertain whether the principle of Fermat’s theorem of the polygonal numbers may not be extended to all orders of series whose ultimate differences are con­stant. The polygonal numbers are all of the quadratic form, and they have (according to Fermat’s theorem) this property, that every number is the sum of not exceeding, 3 terms of the triangular num­bers, 4 of the square numbers, 5 of the pentagonal numbers, &c. It is stated in this paper that the series of the odd squares 1,9,25,49, &c. has a similar property, and that every number is the sum of not exceeding 10 odd squares. It is also stated, that a series con­sisting of the 1st and every succeeding 3rd term of the triangular series, viz. 1,10,28,35, &c., has a similar property; and that every number is the sum of not exceeding 11 terms of this last series, and that this may be easily proved [it was proved in a former paper by the same author]. The term “Notation-limit” is applied to the num­ber which denotes the largest number of terms of a series necessary to express any number; and the writer states that 5,7,9,13,21 are respectively the notation-limits of the tetrahedral numbers, the octa­hedral, the cubical, the eicosahedral and the dodecahedral numbers; that 19 is the notation-limit of the series of the 4th powers; that 11 is the notation-limit of the series of the triangular numbers squared, viz. 1,9,36,100, &c., and 31 the notation-limit of the series 1,28,153, &c. (the sum of the odd cubes), whose general expression is 2 n 4 — n 2 .

scholarly journals Cassini- and Catalan-like formulas for polygonal numbers and simplex numbers

SURG Journal ◽  
2012 ◽  
Vol 5 (2) ◽  
pp. 37-43
Author(s):  
Thomas Jeffery
Keyword(s):  
Fibonacci Numbers ◽  
Triangular Numbers ◽  
Polygonal Numbers

Cassini’s formula and Catalan’s formula are two results from the theory of Fibonacci numbers. This article derives results similar to these, however instead of applying to Fibonacci numbers, they are applied to polygonal numbers and simplex numbers. Triangular numbers are considered first. We then generalize to polygonal and simplex numbers. For polygonal numbers the properties of determinants are used to simplify calculations. For simplex numbers Pascal’s Theorem is used.

A sum of three nonzero triangular numbers

2021 ◽  
pp. 1-22
Author(s):  
Daejun Kim ◽  
Jeongwon Lee ◽  
Byeong-Kweon Oh
Keyword(s):  
Riemann Hypothesis ◽  
Generalized Riemann Hypothesis ◽  
Triangular Numbers ◽  
Finite Set ◽  
Polygonal Numbers ◽  
Sums Of Three Squares ◽  
Number Of Authors

Finding all integers which can be written as a sum of three nonzero squares of integers has been studied by a number of authors. This question is solved under the assumption of the Generalized Riemann Hypothesis (GRH), but still remains unsolved unconditionally. In this paper, we show that out of all integers that are sums of three squares, all but finitely many can be written as [Formula: see text] for some integers [Formula: see text]. Furthermore, we explicitly describe this finite set under the GRH. From this result, we also describe further generalizations for sums of nonzero polygonal numbers. Precisely, we find all integers, under the GRH only when [Formula: see text], which are sums of [Formula: see text] nonzero triangular (generalized pentagonal and generalized octagonal, respectively) numbers for any integer [Formula: see text].

Polygonal numbers: a study of patterns

1970 ◽  
Vol 17 (1) ◽  
pp. 33-38
Author(s):  
Margaret A. Hervey ◽  
Bonnie H. Litwiller
Keyword(s):  
Geometrical Pattern ◽  
Regular Triangle ◽  
Triangular Numbers ◽  
Triangular Number ◽  
Polygonal Numbers

Number patterns have interested mathematicians from the time of the ancient Greeks. Some of this interest centered around the relations between number expressed geometrically and algebraically. This idea is basic to the study of polygonal numbers. Polygonal numbers, which are sometimes called figurate numbers, include triangular numbers, square numbers. pentagonal numbers, hexagonal numbers, and so on. For example, if a number of objects can be arranged in the form of a regular triangle, that number is considered to be a triangular number. Figure I shows a geometrical pattern for the first five triangular numbers.

scholarly journals A DIOPHANTINE PROBLEM CONCERNING POLYGONAL NUMBERS

2013 ◽  
Vol 88 (2) ◽  
pp. 345-350
Author(s):  
DAEYEOUL KIM ◽  
YOON KYUNG PARK ◽  
ÁKOS PINTÉR
Keyword(s):  
Diophantine Problem ◽  
Triangular Numbers ◽  
Polygonal Numbers

AbstractMotivated by some earlier Diophantine works on triangular numbers by Ljunggren and Cassels, we consider similar problems for general polygonal numbers.

scholarly journals XXV. On Fermat's theorem of the polygonal numbers. —Second communication

1863 ◽  
Vol 12 ◽  
pp. 205-209
Keyword(s):  
The Subject ◽  
Polygonal Numbers

The object of this paper is to show the result of combining the three series (which have been the subject of previous communications) in a square, in such manner that the division into 4 squares of certain terms in each series, may produce a division into 4 squares of every term of other series, and thus each term in the whole square will at last be divided into 4 squares, and the first term will be so divided into 4 square numbers that two of the roots will be equal to each other; two of them will differ by 1, and the algebraic sum of all the roots will be equal to 1. It is not offered (at present) as a proof that it must be so, but as a method by which that result may always (in fact) be obtained.

scholarly journals When are Multiples of Polygonal Numbers again Polygonal Numbers?

2019 ◽  
Author(s):  
Jasbir Chahal ◽  
Michael Griffin ◽  
Nathan Priddis
Keyword(s):  
Fundamental Unit ◽  
Pell Equation ◽  
Existence Of A Solution ◽  
Easy Consequence ◽  
Triangular Numbers ◽  
International Audience ◽  
Polygonal Numbers ◽  
Erratic Behavior

International audience Euler showed that there are infinitely many triangular numbers that are three times other triangular numbers. In general, it is an easy consequence of the Pell equation that for a given square-free m > 1, the relation ∆ = m∆' is satisfied by infinitely many pairs of triangular numbers ∆, ∆'. After recalling what is known about triangular numbers, we shall study this problem for higher polygonal numbers. Whereas there are always infinitely many triangular numbers which are fixed multiples of other triangular numbers, we give an example that this is false for higher polygonal numbers. However, as we will show, if there is one such solution, there are infinitely many. We will give conditions which conjecturally assure the existence of a solution. But due to the erratic behavior of the fundamental unit of Q(√ m), finding such a solution is exceedingly difficult. Finally, we also show in this paper that, given m > n > 1 with obvious exceptions, the system of simultaneous relations P = mP' , P = nP'' has only finitely many possibilities not just for triangular numbers, but for triplets P , P' , P'' of polygonal numbers, and give examples of such solutions.

Triangular Numbers: The Building Blocks of Figurate Numbers

1983 ◽  
Vol 76 (8) ◽  
pp. 624-625
Author(s):  
Melfried Olson ◽  
Gerald K. Goff ◽  
Murray Blose
Keyword(s):  
Mathematics Education ◽  
Building Blocks ◽  
Triangular Numbers ◽  
Polygonal Numbers

Figurate or polygonal numbers have been studied since the Greek era. Recent publications in mathematics education have reflected their continuing attraction (Hartman 1976; Smith 1972; Weaver 1974).

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