[2403.00760] Faulhaber's formula, Bernoulli numbers, power sums of natural numbers and totatives and the functional equation $f(x)+x^k=f(x+1)$ Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:2403.00760 (math) [Submitted on 1 Mar 2024 ( v1 ), last revised 29 Jun 2026 (this version, v4)] Title: Faulhaber's formula, Bernoulli numbers, power sums of natural numbers and totatives and the functional equation $f(x)+x^k=f(x+1)$ Authors: Chai Wah Wu View a PDF of the paper titled Faulhaber's formula, Bernoulli numbers, power sums of natural numbers and totatives and the functional equation $f(x)+x^k=f(x+1)$, by Chai Wah Wu View PDF HTML (experimental) Abstract: In modern usage the Bernoulli numbers and Bernoulli polynomials follow Euler's approach and are defined using generating functions. Originally, they were derived by Bernoulli while characterizing Faulhaber's formula for the sum of consecutive powers. These equations have many consequences and applications in various areas of mathematics. We consider yet another application by studying the functional equation $f(x)+x^k=f(x+1)$ and show that a solution of this equation can be derived from Faulhaber's formula. We then use these results to study the totatives of n, i.e. numbers less than n that are coprime to n. In particular, we look at sums of powers of totatives of n that are less than n/2. We show that the sum of powers of this half of the totatives can also be expressed in the same structural form as the sum of powers of all totatives and provide explicit formulas for this sum. As an application of these results, we obtain a formula for the total area of all rectangles with coprime width and length and semiperimeter n. Comments: 10 pages Subjects: Number Theory (math.NT) ; Combinatorics (math.CO); History and Overview (math.HO) MSC classes: 11B68, 11A05 Cite as: arXiv:2403.00760 [math.NT] (or arXiv:2403.00760v4 [math.NT] for this version) https://doi.org/10.48550/arXiv.2403.00760 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Chai Wah Wu [ view email ] [v1] Fri, 1 Mar 2024 18:58:32 UTC (4 KB) [v2] Sat, 25 Apr 2026 02:19:42 UTC (9 KB) [v3] Tue, 28 Apr 2026 23:28:48 UTC (9 KB) [v4] Mon, 29 Jun 2026 21:38:40 UTC (10 KB) Full-text links: Access Paper: View a PDF of the paper titled Faulhaber's formula, Bernoulli numbers, power sums of natural numbers and totatives and the functional equation $f(x)+x^k=f(x+1)$, by Chai Wah Wu View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2024-03 Change to browse by: math math.CO math.HO References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from