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].

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 Some New Notes on Mersenne Primes and Perfect Numbers

2020 ◽  
Vol 3 (1) ◽  
pp. 15
Author(s):  
Leomarich F Casinillo
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Mathematical Structure ◽  
Prime Numbers ◽  
Perfect Number ◽  
Positive Divisor ◽  
Triangular Numbers ◽  
Perfect Numbers ◽  
Mersenne Primes ◽  
Mersenne Prime

<p>Mersenne primes are specific type of prime numbers that can be derived using the formula <img title="\large M_p=2^{p}-1" src="https://latex.codecogs.com/gif.latex?\large&amp;space;M_p=2^{p}-1" alt="" />, where <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&amp;space;p" alt="" /> is a prime number. A perfect number is a positive integer of the form <img title="\large P(p)=2^{p-1}(2^{p}-1)" src="https://latex.codecogs.com/gif.latex?\large&amp;space;P(p)=2^{p-1}(2^{p}-1)" alt="" /> where <img title="\large 2^{p}-1" src="https://latex.codecogs.com/gif.latex?\large&amp;space;2^{p}-1" alt="" /> is prime and <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&amp;space;p" alt="" /> is a Mersenne prime, and that can be written as the sum of its proper divisor, that is, a number that is half the sum of all of its positive divisor. In this note, some concepts relating to Mersenne primes and perfect numbers were revisited. Further, Mersenne primes and perfect numbers were evaluated using triangular numbers. This note also discussed how to partition perfect numbers into odd cubes for odd prime <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&amp;space;p" alt="" />. Also, the formula that partition perfect numbers in terms of its proper divisors were constructed and determine the number of primes in the partition and discuss some concepts. The results of this study is useful to better understand the mathematical structure of Mersenne primes and perfect numbers.</p>

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.

Pseudoperfect numbers with no small prime divisors

2009 ◽  
Vol 93 (528) ◽  
pp. 404-409
Author(s):  
Peter Shiu
Keyword(s):  
Difficult Problem ◽  
Prime Divisors ◽  
Perfect Number ◽  
Perfect Numbers ◽  
Mersenne Primes ◽  
Mersenne Prime

A perfect number is a number which is the sum of all its divisors except itself, the smallest such number being 6. By results due to Euclid and Euler, all the even perfect numbers are of the form 2P-1(2p - 1) where p and 2p - 1 are primes; the latter one is called a Mersenne prime. Whether there are infinitely many Mersenne primes is a notoriously difficult problem, as is the problem of whether there is an odd perfect number.

scholarly journals Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes

10.37236/1052 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Brad Jackson ◽  
Frank Ruskey
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Binary Trees ◽  
Integer Sequences ◽  
Number Of Leaves ◽  
Online Encyclopedia ◽  
Fibonacci Sequences ◽  
Infinite Sequences

We consider a family of meta-Fibonacci sequences which arise in studying the number of leaves at the largest level in certain infinite sequences of binary trees, restricted compositions of an integer, and binary compact codes. For this family of meta-Fibonacci sequences and two families of related sequences we derive ordinary generating functions and recurrence relations. Included in these families of sequences are several well-known sequences in the Online Encyclopedia of Integer Sequences (OEIS).

Digital Sums of Perfect Numbers and Triangular Numbers

1969 ◽  
Vol 62 (3) ◽  
pp. 179-182
Author(s):  
Robert W. Prielipp
Keyword(s):  
Number Theory ◽  
Elementary Level ◽  
Present Evidence ◽  
Triangular Numbers ◽  
Perfect Numbers

BEFORE a theorem can be proved it must first be discovered. One of the areas of mathematics in which some very intriguing results can be uncovered, even at a relatively elementary level, is that of number theory. We proceed to present evidence to substantiate the preceding remark.

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.

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