Pollock's conjectures

Last updated

Pollock's conjectures are closely related conjectures in additive number theory. [1] They were first stated in 1850 by Sir Frederick Pollock, [1] [2] better known as a lawyer and politician, but also a contributor of papers on mathematics to the Royal Society. These conjectures are a partial extension of the Fermat polygonal number theorem to three-dimensional figurate numbers, also called polyhedral numbers.

Statement of the conjectures

The numbers that are not the sum of at most 4 tetrahedral numbers are given by the sequence 17, 27, 33, 52, 73, ..., (sequence A000797 in the OEIS ) of 241 terms, with 343,867 conjectured to be the last such number. [3]

This conjecture has been proven for all but finitely many positive integers. [4]

The cube numbers case was established from 1909 to 1912 by Wieferich [5] and A. J. Kempner. [6]

This conjecture was confirmed as true in 2023. [7]

Related Research Articles

<span class="mw-page-title-main">Prime number</span> Number divisible only by 1 or itself

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.

In number theory, Waring's problem asks whether each natural number k has an associated positive integer s such that every natural number is the sum of at most s natural numbers raised to the power k. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. Waring's problem was proposed in 1770 by Edward Waring, after whom it is named. Its affirmative answer, known as the Hilbert–Waring theorem, was provided by Hilbert in 1909. Waring's problem has its own Mathematics Subject Classification, 11P05, "Waring's problem and variants".

<span class="mw-page-title-main">Goldbach's conjecture</span> Even integers as sums of two primes

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers.

In mathematics, a Fermat number, named after Pierre de Fermat, the first known to have studied them, is a positive integer of the form: where n is a non-negative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ....

<i>abc</i> conjecture The product of distinct prime factors of a,b,c, where c is a+b, is rarely much less than c

The abc conjecture is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985. It is stated in terms of three positive integers and that are relatively prime and satisfy . The conjecture essentially states that the product of the distinct prime factors of is usually not much smaller than . A number of famous conjectures and theorems in number theory would follow immediately from the abc conjecture or its versions. Mathematician Dorian Goldfeld described the abc conjecture as "The most important unsolved problem in Diophantine analysis".

The modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way. Andrew Wiles and Richard Taylor proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem. Later, a series of papers by Wiles's former students Brian Conrad, Fred Diamond and Richard Taylor, culminating in a joint paper with Christophe Breuil, extended Wiles's techniques to prove the full modularity theorem in 2001.

In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians.

4000 is the natural number following 3999 and preceding 4001. It is a decagonal number.

<span class="mw-page-title-main">Octahedral number</span>

In number theory, an octahedral number is a figurate number that represents the number of spheres in an octahedron formed from close-packed spheres. The th octahedral number can be obtained by the formula:

In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most nn-gonal numbers. That is, every positive integer can be written as the sum of three or fewer triangular numbers, and as the sum of four or fewer square numbers, and as the sum of five or fewer pentagonal numbers, and so on. That is, the n-gonal numbers form an additive basis of order n.

In mathematics, Wolstenholme's theorem states that for a prime number , the congruence

<span class="mw-page-title-main">Centered nonagonal number</span> Centered figurate number that represents a nonagon with a dot in the center

A centered nonagonal number is a centered figurate number that represents a nonagon with a dot in the center and all other dots surrounding the center dot in successive nonagonal layers. The centered nonagonal number for n layers is given by the formula

Arthur Josef Alwin Wieferich was a German mathematician and teacher, remembered for his work on number theory, as exemplified by a type of prime numbers named after him.

Markov number or Markoff number is a positive integer x, y or z that is part of a solution to the Markov Diophantine equation

<span class="mw-page-title-main">5</span> Integer number 5

5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number. It has garnered attention throughout history in part because distal extremities in humans typically contain five digits.

In additive number theory, an additive basis is a set of natural numbers with the property that, for some finite number , every natural number can be expressed as a sum of or fewer elements of . That is, the sumset of copies of consists of all natural numbers. The order or degree of an additive basis is the number . When the context of additive number theory is clear, an additive basis may simply be called a basis. An asymptotic additive basis is a set for which all but finitely many natural numbers can be expressed as a sum of or fewer elements of .

In mathematics and statistics, sums of powers occur in a number of contexts:

<span class="mw-page-title-main">Fermat's Last Theorem</span> 17th-century conjecture proved by Andrew Wiles in 1994

In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions.

In number theory, the Fermat quotient of an integer a with respect to an odd prime p is defined as

References

  1. 1 2 Dickson, L. E. (June 7, 2005). History of the Theory of Numbers, Vol. II: Diophantine Analysis. Dover. pp. 22–23. ISBN   0-486-44233-0.
  2. Frederick Pollock (1850). "On the extension of the principle of Fermat's theorem on the polygonal numbers to the higher order of series whose ultimate differences are constant. With a new theorem proposed, applicable to all the orders". Abstracts of the Papers Communicated to the Royal Society of London. 5: 922–924. JSTOR   111069.
  3. Weisstein, Eric W. "Pollock's Conjecture". MathWorld .
  4. Elessar Brady, Zarathustra (2016). "Sums of seven octahedral numbers". Journal of the London Mathematical Society. Second Series. 93 (1): 244–272. arXiv: 1509.04316 . doi:10.1112/jlms/jdv061. MR   3455791. S2CID   206364502.
  5. Wieferich, Arthur (1909). "Beweis des Satzes, daß sich eine jede ganze Zahl als Summe von höchstens neun positiven Kuben darstellen läßt". Mathematische Annalen (in German). 66 (1): 95–101. doi:10.1007/BF01450913. S2CID   121386035.
  6. Kempner, Aubrey (1912). "Bemerkungen zum Waringschen Problem". Mathematische Annalen (in German). 72 (3): 387–399. doi:10.1007/BF01456723. S2CID   120101223.
  7. Kureš, Miroslav (2023-10-27). "A Proof of Pollock's Conjecture on Centered Nonagonal Numbers". The Mathematical Intelligencer. doi:10.1007/s00283-023-10307-0. ISSN   0343-6993.