SOME OBSERVATIONS AND SPECULATIONS ON PARTITIONS INTO <i>d</i>-TH POWERS

نویسندگان

چکیده

Abstract The aim of this article is to provoke discussion concerning arithmetic properties the function $p_{d}(n)$ counting partitions a positive integer n into d th powers, where $d\geq 2$ . Apart from results asymptotic behaviour , little known. In first part paper, we prove certain congruences involving functions various types powers. second paper experimental and contains questions conjectures sequence $(p_{d}(n))_{n\in \mathbb {N}}$ based on computations for $n\leq 10^5$ $d=2$ 10^{6}$ $d=3, 4, 5$

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Partitions of an Integer into Powers

In this paper, we use a simple discrete dynamical model to study partitions of integers into powers of another integer. We extend and generalize some known results about their enumeration and counting, and we give new structural results. In particular, we show that the set of these partitions can be ordered in a natural way which gives the distributive lattice structure to this set. We also giv...

متن کامل

Some Observations on Dyson's New Symmetries of Partitions

We utilize Dyson’s concept of the adjoint of a partition to derive an infinite family of new polynomial analogues of Euler’s Pentagonal Number Theorem. We streamline Dyson’s bijection relating partitions with crank ≤ k and those with k in the Rank-Set of partitions. Also, we extend Dyson’s adjoint of a partition to MacMahon’s “modular” partitions with modulus 2. This way we find a new combinato...

متن کامل

Some formulae for partitions into squares

We consider the problem of determining the number of partitions of a number into a fixed number of squares. This problem is distinct from, but not unrelated to, the corresponding classical problem of determining the number of representations of a number as the sum of a fixed number of squares. Whereas the classical problem has received an enormous amount of attention over the years, see Dickson...

متن کامل

Strange Quark stars: Observations & Speculations

Two kinds of difficulties have challenged the physics community for many years: (1) knowing nature’s building blocks (particle physics) and (2) understanding interacting many-body systems (many-body physics). Both of them exist in the research of quark matter and compact stars. This paper addresses the possibility that quark clustering, rather than a color super-conducting state, could occur in...

متن کامل

Some speculations on the myeloproliferative syndromes.

Some Speculations on the Myeloproliferative Syndromes W ITH ACCUMULATING EXI ERIENCE, it becomes more and more evident that the bone marrow cells-enythrobhasts, granulocytes, megakar-vocvtes-often proliferate en masse or as a unit i-ather than as single elements. Thus, heukocytosis an(h thIOml)OcytOsis, as well as reticuhocytosis occur not only in 1)enign reversible cond!itions such as hemorrha...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

ژورنال

عنوان ژورنال: Bulletin of The Australian Mathematical Society

سال: 2021

ISSN: ['0004-9727', '1755-1633']

DOI: https://doi.org/10.1017/s0004972721000034