Pollock's conjectures are two closely related unproven

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...

s in

additive number theory Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of abelian groups and commutative semigr ...

. They were first stated in 1850 by Sir Frederick Pollock, better known as a lawyer and politician, but also a contributor of papers on mathematics to the

Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...

. These conjectures are a partial extension of the

Fermat polygonal number theorem In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most -gonal numbers. That is, every positive integer can be written as the sum of three or fewer triangular numbers, and as the sum ...

to three-dimensional

figurate numbers The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polygon ...

, also called polyhedral numbers. *Pollock tetrahedral numbers conjecture: Every positive integer is the sum of at most five

tetrahedral number A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The th tetrahedral number, , is the sum of the first triangular numbers, that is, ...

s. The numbers that are not the sum of at most 4 tetrahedral numbers are given by the sequence 17, 27, 33, 52, 73, ..., of 241 terms, with 343867 being almost certainly the last such number. *Pollock octahedral numbers conjecture: Every positive integer is the sum of at most seven

octahedral number 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 ''n''th octahedral number O_n can be obtained by the formula:. :O_n=. The first few octahed ...

s. This conjecture has been proven for all but finitely many positive integers. *Polyhedral numbers conjecture: Let ''m'' be the number of vertices of a

platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...

“regular ''n''-hedron” (''n'' is 4, 6, 8, 12, or 20), then every positive integer is the sum of at most ''m''+1 ''n''-hedral numbers. (i.e. every positive integer is the sum of at most 5

s, or the sum of at most 9 cube numbers, or the sum of at most 7

s, or the sum of at most 21 dodecahedral numbers, or the sum of at most 13 icosahedral numbers)

References

Conjectures Unsolved problems in number theory Figurate numbers Additive number theory {{numtheory-stub