{"id":"0792897f-0448-4d47-aecc-1a688c267676","title":"Liouville's Divisor Sum Theorem Proof","created_at":"2025-12-05T05:46:15Z","saved":true,"shared":true,"branch_id":"00000000-0000-4000-0000-000000000000","profile":{"id":null,"name":null,"subtitle":"State of the art deep research (>30 secs)","model":"ki_research","model_name":"Research","model_provider":"kagi","model_input_limit":1000000,"recommended":false,"model_info":"\n\n<div class=\"heading\">\n <div class=\"left\">\n \n \n \n <svg class=\"model-icon icon-xs\" data-model=\"ki_research\">\n <use href=\"#icon-kagi\"/>\n </svg>\n <h3>\n \n <span class=\"model-provider\">Kagi</span>\n \n <span class=\"model-name\">none</span>\n </h3>\n </div>\n\n \n <a href=\"/settings?p=custom_assistant&id=none\">\n <svg class=\"icon-sm edit-icon\" aria-hidden=\"true\">\n <use href=\"#icon-pencil\" />\n </svg>\n </a>\n \n</div>\n\n\n<hr />\n<ul>\n \n <li>\n <h4>\n Cost\n </h4>\n <span>\n \n \n \n\n \n \n \n\n <div class=\"score\" aria-label=\"Expensive\">\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-cost --high\"></div>\n \n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-cost --high\"></div>\n \n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-cost --high\"></div>\n \n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-cost --high\"></div>\n \n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-cost --high\"></div>\n \n \n </div>\n</span>\n </li>\n <li>\n <h4>\n Quality\n </h4>\n <span>\n \n \n \n\n \n \n \n\n <div class=\"score\" aria-label=\"Rated 5 out of 5\">\n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n </div>\n</span>\n </li>\n <li>\n <h4>\n Speed\n </h4>\n <span>\n \n \n \n\n \n \n \n\n <div class=\"score\" aria-label=\"Rated 1 out of 5\">\n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n </div>\n</span>\n </li>\n <li>\n <h4>\n Privacy\n </h4>\n <span>\n \n \n \n\n \n \n \n\n <div class=\"score\" aria-label=\"Rated 4 out of 5\">\n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n </div>\n</span>\n </li>\n <li class=\"divider\" aria-hidden=\"true\"></li>\n <li>\n <h4>Last updated</h4>\n <span class=\"date\">\n \n \n 2025-11-18\n \n \n </span>\n </li>\n \n</ul>\n","model_provider_name":"Kagi","internet_access":true,"personalizations":true,"shortcut":null,"is_default_profile":false},"tag_ids":[],"read_only":true}

[{"id":"999b5eeb-9e1b-4d6a-81f4-624b0a032762","thread_id":"0792897f-0448-4d47-aecc-1a688c267676","created_at":"2025-12-05T05:46:15Z","state":"done","prompt":"Prove Liousville’s divisor sum theorem that assets the squared sum of the number of divisors of each divisor of a number is equal to the sum of the cubes of the same numbers","references_html":"<ol data-ref-list>\n \n <li>\n <a href=\"https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/generalising-sums-of-cubes-equal-to-squares-of-sums/064A51DC4FAB6AEA9BCCC21A7ED62A70\" >Generalising ‘Sums of cubes equal to squares of sums’ | The Mathematical Gazette | Cambridge Core</a>\n <span class=\"__domain-name\">www.cambridge.org</span>\n <div class=\"trigger\" tabindex=\"0\">\n <span class=\"contribution\" style=\"background: linear-gradient(to right, var(--primary-70) 65%, transparent 65%)\">65%</span>\n <div class=\"tooltip top end tooltip_base tw-auto\">\n This percentage indicates how much a source has contributed to the answer.\n </div>\n </div>\n </li>\n \n <li>\n <a href=\"https://nntdm.net/papers/nntdm-19/NNTDM-19-1-01-13.pdf\" >NNTDM-19-1-01-13.pdf</a>\n <span class=\"__domain-name\">nntdm.net</span>\n <div class=\"trigger\" tabindex=\"0\">\n <span class=\"contribution\" style=\"background: linear-gradient(to right, var(--primary-70) 35%, transparent 35%)\">35%</span>\n <div class=\"tooltip top end tooltip_base tw-auto\">\n This percentage indicates how much a source has contributed to the answer.\n </div>\n </div>\n </li>\n \n</ol>\n","references_md":"[^1]: [Generalising ‘Sums of cubes equal to squares of sums’ | The Mathematical Gazette | Cambridge Core](https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/generalising-sums-of-cubes-equal-to-squares-of-sums/064A51DC4FAB6AEA9BCCC21A7ED62A70) (65%)\n[^2]: [NNTDM-19-1-01-13.pdf](https://nntdm.net/papers/nntdm-19/NNTDM-19-1-01-13.pdf) (35%)\n","reply":"<details><summary>Planned research</summary>\n<ul>\n<li>Search for Liouville's divisor sum theorem statement and proof</li>\n<li>Verify the exact mathematical formulation</li>\n<li>Find detailed mathematical proof or derivation</li>\n</ul>\n</details>\n<details><summary>Searched with Kagi<p></p>\n<ul>\n<li>Liouville divisor sum theorem proof squared sum divisors cubes</li></ul></summary><br>\nSources (5):<br>\n<strong>Sum of Cubes is Square of Sum... And More! - Pat'sBlog</strong><br>\n<a href=\"https://pballew.blogspot.com/2018/06/sum-of-cubes-is-square-of-sum-and-more.html\" target=\"_blank\">https://pballew.blogspot.com/2018/06/sum-of-cubes-is-square-of-sum-and-more.html</a><br>\nJun 15, 2018 I also discovered that the sums of sets with this Liouville property had a multiplication rule similar to the ones for finding the number of divisors....First, Primes to any power (even zero power) always have one more divisor than their exponent, so 32 has three divisors, 1, 3, and 9. , so if you wonder about numbers like 18, which is 32 * 2, you can multiply the three divisors of 9, by the two divisors of 2 , to find that 18 has six total divisors. Now the same kind of properties work with the sums of the Liouville sets.\n\n<p><strong>Sums of Cubes as Square of Sum, with a Why? and More. - Pat'sBlog</strong><br>\n<a href=\"https://pballew.blogspot.com/2022/06/sums-of-cubes-as-square-of-sum-with-why.html\" target=\"_blank\">https://pballew.blogspot.com/2022/06/sums-of-cubes-as-square-of-sum-with-why.html</a><br>\nJun 23, 2022 Now let's make a set of these four new numbers, {1,2,2,4} What Liouville discovered was that the sum of the cubes of these four numbers, ( (1^3 ...</p>\n<p><strong>SumofCubesisSquareofSum Sum o - arXiv.org</strong><br>\n<a href=\"https://arxiv.org/pdf/1306.5257.pdf\" target=\"_blank\">https://arxiv.org/pdf/1306.5257.pdf</a><br>\n SumofCubesisSquareofSum Sum of Cubes is Square of Sum 2 Edward Barbeau 1, Samer Seraj University of Toronto Abstract. Inspired by the fact that the sum of the cubes of the first n naturals is equal to the square of their sum, we explore, for each n, the Diophantine equation representing all non-trivial sets of n integers with this property. We find definite answers to the standard question of ...</p>\n<p><strong>Sums of Cubes as Square of Sum, with a Why? and More.</strong><br>\n<a href=\"https://pballew.blogspot.com/2021/06/sums-of-cubes-as-square-of-sum-with-why.html\" target=\"_blank\">https://pballew.blogspot.com/2021/06/sums-of-cubes-as-square-of-sum-with-why.html</a><br>\nJun 01, 2021 The cubes sum to 729 and the sum (27) also squares to 729. And if you walk through the process of producing the Liouville set for 3*10 = 30, you will find it matches....but then so would 105, or in general, any sphenic number, the product of three distinct primes.</p>\n<p><strong>Divisor function - Wikipedia</strong><br>\n<a href=\"https://en.wikipedia.org/wiki/Divisor_function\" target=\"_blank\">https://en.wikipedia.org/wiki/Divisor_function</a><br>\nDec 26, 2008 Divisor function σ0 (n) up to n = 250 Sigma function σ1 (n) up to n = 250 Sum of the squares of divisors, σ2 (n), up to n = 250 Sum of cubes of divisors, σ3 (n) up to n = 250 In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an ...</p>\n</details>\n<details><summary>Contemplated</summary><br>\n The search results mention Liouville's property and sum of cubes being square of sum, but I need to find the specific theorem statement about divisors. The user's description seems to be about a specific divisor theorem. Let me search more specifically for Liouville's theorem about divisors and their relationship.<p></p>\n</details>\n<details><summary>Searched with Kagi<p></p>\n<ul>\n<li>Liouville theorem divisors sum squared cubes mathematical proof</li>\n<li>sum of squares of number of divisors equals sum of cubes divisors</li></ul></summary><br>\nSources (10):<br>\n<strong>Sum of Cubes is Square of Sum... And More! - Pat'sBlog</strong><br>\n<a href=\"https://pballew.blogspot.com/2018/06/sum-of-cubes-is-square-of-sum-and-more.html\" target=\"_blank\">https://pballew.blogspot.com/2018/06/sum-of-cubes-is-square-of-sum-and-more.html</a><br>\nJun 15, 2018 The mathematical (and other) thoughts of a (now retired) math teacherWhen I played around with the idea, I realized that all the strings of natural numbers (powers of a prime) summed to a triangular number (64 above sums to 28, the 7th triangular number), but no others. Semiprimes always produce {1,2,2,4} which has a sum of nine, I also discovered that the sums of sets with this Liouville property had a multiplication rule similar to the ones for finding the number of divisors....\n\n<p><strong>Generalising ‘Sums of cubes equal to squares of sums’</strong><br>\n<a href=\"https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/generalising-sums-of-cubes-equal-to-squares-of-sums/064A51DC4FAB6AEA9BCCC21A7ED62A70\" target=\"_blank\">https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/generalising-sums-of-cubes-equal-to-squares-of-sums/064A51DC4FAB6AEA9BCCC21A7ED62A70</a><br>\n David Pagni drew attention to a result which is ascribed by Dickson [2, p. 286] to Liouville (1857), that the sum of the cubes of the number of divisors of each of the divisors of an integer, is equal to the square of their sum. For example, the divisors of 6 are 1, 2, 3, and 6, which have 1, 2, 2, and 4 divisors respectively, and Pagni observed, as have others, including Mason et al. [4, p ...</p>\n<p><strong>Liouville function - Wikipedia</strong><br>\n<a href=\"https://en.wikipedia.org/wiki/Liouville_function\" target=\"_blank\">https://en.wikipedia.org/wiki/Liouville_function</a><br>\n The sum of the Liouville function over the divisors of n {\\displaystyle n}. {\\displaystyle n}. is the characteristic function of the squares: ∑ d | n λ ( d ) ...</p>\n<p><strong>Liouville's theorem (complex analysis) - Wikipedia</strong><br>\n<a href=\"https://en.wikipedia.org/wiki/Liouville%27s_theorem_(complex_analysis\" target=\"_blank\">https://en.wikipedia.org/wiki/Liouville%27s_theorem_(complex_analysis)</a>)<br>\nDec 26, 2008 Proof (Fundamental theorem of algebra). Suppose for the sake of contradiction that there is a nonconstant polynomial.^ Cauchy, Augustin-Louis (1844), \"Mémoires sur les fonctions complémentaires\", Œuvres complètes d'Augustin Cauchy, 1, vol. 8, Paris: Gauthiers-Villars (published 1882). ^ Lützen, Jesper (1990), Joseph Liouville 1809–1882: Master of Pure and Applied Mathematics, Studies in the History of Mathematics and Physical Sciences, vol. 15, Springer-Verlag, ISBN 3-540-97180-7.</p>\n<p><strong>Complex Analysis Proof Application of Liouville's Theorem</strong><br>\n<a href=\"https://math.stackexchange.com/questions/2217272/complex-analysis-proof-application-of-liouvilles-theorem\" target=\"_blank\">https://math.stackexchange.com/questions/2217272/complex-analysis-proof-application-of-liouvilles-theorem</a><br>\nApr 25, 2017 My professor gave a hint to use the power series expansion of <math display=\"inline\"><mi>f</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>z</mi><mo symmetric=\"false\" stretchy=\"false\">)</mo></math> (to manipulate <math display=\"inline\"><mi>f</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>z</mi><mo symmetric=\"false\" stretchy=\"false\">)</mo></math>) and then to use or apply Liouville's Theorem. I have tried this, but am getting nowhere. Any help, suggestions, or tips would be much appreciated. Please note that this is for an elementary course on complex analysis so please do not give too advanced an explanation. Thank you. I have attached two pictures, one of the Cauchy estimates and one of the actual question. enter image description here.</p>\n<p><strong>Divisor function - Wikipedia</strong><br>\n<a href=\"https://en.wikipedia.org/wiki/Divisor_function\" target=\"_blank\">https://en.wikipedia.org/wiki/Divisor_function</a><br>\nDec 26, 2008 Sum of cubes of divisors, σ3(n) up to n = 250. In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer.Divisor sum convolutions, lists a few identities involving the divisor functions. Euler's totient function, Euler's phi function. Refactorable number. Table of divisors. Unitary divisor.</p>\n<p><strong>Arithmetic function - Wikipedia</strong><br>\n<a href=\"https://en.wikipedia.org/wiki/Arithmetic_function\" target=\"_blank\">https://en.wikipedia.org/wiki/Arithmetic_function</a><br>\nDec 26, 2008 In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function whose domain is the set of positive integers and whose range is a subset of the complex numbers. Hardy &amp; Wright include in their definition the requirement that an arithmetical function \"expresses some arithmetical property of n\". There is a larger class of number-theoretic functions that do not fit this definition, for example, the prime-counting functions.</p>\n<p><strong>Sum of squares function - Wikipedia</strong><br>\n<a href=\"https://en.wikipedia.org/wiki/Sum_of_squares_function\" target=\"_blank\">https://en.wikipedia.org/wiki/Sum_of_squares_function</a><br>\nMar 17, 2016 Integers satisfying the sum of two squares theorem are squares of possible distances between integer lattice points; values up to 100 are shown, with. • Squares (and thus integer distances) in red. • Non-unique representations (up to rotation and reflection) bolded.The number of ways to represent n as the sum of four squares was due to Carl Gustav Jakob Jacobi and it is eight times the sum of all its divisors which are not divisible by 4, i.e.</p>\n<p><strong>Square number - Wikipedia</strong><br>\n<a href=\"https://en.wikipedia.org/wiki/Square_number\" target=\"_blank\">https://en.wikipedia.org/wiki/Square_number</a><br>\n Square number Square number 16 as sum of gnomons. In mathematics, a square number or perfect square is an integer that is the square of an integer; [1] in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals 32 and can be written as 3 × 3.</p>\n<p><strong>Fibonacci sequence - Wikipedia</strong><br>\n<a href=\"https://en.wikipedia.org/wiki/Fibonacci_sequence\" target=\"_blank\">https://en.wikipedia.org/wiki/Fibonacci_sequence</a><br>\n In mathematics, the Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted Fn .</p>\n</details>\n<details><summary>Contemplated</summary><br>\n Great! I found the exact theorem in citation 6. It states that Liouville (1857) proved \"the sum of the cubes of the number of divisors of each of the divisors of an integer, is equal to the square of their sum.\" This matches the user's description. Let me get more details about this theorem and its proof.<p></p>\n</details>\n<details><summary>Gathered key details from:<p></p>\n<ul>\n<li><a href=\"https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/generalising-sums-of-cubes-equal-to-squares-of-sums/064A51DC4FAB6AEA9BCCC21A7ED62A70\" target=\"_blank\">Generalising ‘Sums of cubes equal to squares of sums’ | The Mathematical Gazette | Cambridge Core</a></li></ul></summary>\n\n<h3>Query</h3>\n<p>What is the exact statement and proof of Liouville's theorem about the sum of cubes of the number of divisors being equal to the square of their sum? Include the mathematical formulation and proof details.</p>\n<h3>Response</h3>\n<p>Using full content.</p>\n</details>\n<details><summary>Contemplated</summary><br>\n I have the theorem statement and an example, but I need to find the actual mathematical proof. The source mentions that it's related to sequences of the form {1, 2, ..., t} having the property that sum of cubes equals square of sum. Let me search for more detailed mathematical proofs or derivations.<p></p>\n</details>\n<details><summary>Searched with Kagi<p></p>\n<ul>\n<li>Liouville 1857 divisor theorem proof mathematical derivation</li>\n<li>sum cubes number divisors square sum proof number theory</li></ul></summary><br>\nSources (10):<br>\n<strong>Liouville's theorem (differential algebra) - Wikipedia</strong><br>\n<a href=\"https://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra\" target=\"_blank\">https://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra)</a>)<br>\nIn mathematics, Liouville's theorem, originally formulated by French mathematician Joseph Liouville in 1833 to 1841, [1][2][3] places an important restriction on antiderivatives that can be expressed as elementary functions.\n\n<p><strong>Liouville's theorem (complex analysis) - Wikipedia</strong><br>\n<a href=\"https://en.wikipedia.org/wiki/Liouville%27s_theorem_(complex_analysis\" target=\"_blank\">https://en.wikipedia.org/wiki/Liouville%27s_theorem_(complex_analysis)</a>)<br>\nDec 26, 2008 Proof (Fundamental theorem of algebra). Suppose for the sake of contradiction that there is a nonconstant polynomial. p {\\displaystyle p}.^ Lützen, Jesper (1990), Joseph Liouville 1809–1882: Master of Pure and Applied Mathematics, Studies in the History of Mathematics and Physical Sciences, vol. 15, Springer-Verlag, ISBN 3-540-97180-7.</p>\n<p><strong>Liouville's theorem (Hamiltonian) - Wikipedia</strong><br>\n<a href=\"https://en.wikipedia.org/wiki/Liouville%27s_theorem_(Hamiltonian\" target=\"_blank\">https://en.wikipedia.org/wiki/Liouville%27s_theorem_(Hamiltonian)</a>)<br>\nMay 05, 2007 In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics.Liouville's theorem applies to conservative systems, that is, systems in which the effects of friction are absent or can be ignored. The general mathematical formulation for such systems is the measure-preserving dynamical system.</p>\n<p><strong>Liouville–Arnold theorem - Wikipedia</strong><br>\n<a href=\"https://en.wikipedia.org/wiki/Liouville%E2%80%93Arnold_theorem\" target=\"_blank\">https://en.wikipedia.org/wiki/Liouville%E2%80%93Arnold_theorem</a><br>\nJul 01, 2015 The Liouville–Arnold theorem is a result in classical mechanics which says, roughly speaking, that seemingly complicated systems can be described as combinations of simple systems if they satisfy certain conditions. The theorem is stated in the language of Hamiltonian mechanics, an approach to mechanics that focuses on the concept of energy. It is named after Joseph Liouville and Vladimir Arnold.</p>\n<p><strong>Liouville</strong><br>\n<a href=\"https://galileoandeinstein.phys.virginia.edu/7010/CM_11_Introduction_to_Liouville.html\" target=\"_blank\">https://galileoandeinstein.phys.virginia.edu/7010/CM_11_Introduction_to_Liouville.html</a><br>\nAug 07, 2019 and Liouville’s theorem is proved. Simpler Proof of Liouville’s Theorem. Landau’s proof given above is extremely elegant: since phase space paths cannot intersect, point inside a volume stay inside, no matter how the volume contorts, and since time development is a canonical transformation, the total volume, given by integrating over volume elements.</p>\n<p>**<a href=\"https://nntdm.net/papers/nntdm-19/NNTDM-19-1-01-13.pdf\" target=\"_blank\">https://nntdm.net/papers/nntdm-19/NNTDM-19-1-01-13.pdf</a>/papers/nntdm-19/NNTDM-19-1-01-13.pdf)<br>\nMar 15, 2017 Notes on Number Theory and Discrete Mathematics Vol. 19, 2013, No. 1, 1–13. Sum of cubes is square of sum. Edward Barbeau and Samer Seraj.Proposition 1. For each natural n, denote by τ (n) the number of positive divisors of n. Then τ (d) : d|n, d ≥ 1 is a CS-set, where τ is taken over the positive divisors of n. Proof. The proposition is clearly true for prime powers since for each prime p and each natural n, the corresponding set of pn−1 is.</p>\n<p><strong>Sum of cubes is square of sum | Notes on Number Theory and Discrete ...</strong><br>\n<a href=\"https://nntdm.net/volume-19-2013/number-1/01-13/\" target=\"_blank\">https://nntdm.net/volume-19-2013/number-1/01-13/</a><br>\n Inspired by the fact that the sum of the cubes of the first n naturals is equal to the square of their sum, we explore, for each n, the Diophantine equation representing all non-trivial sets of n integers with this property. We find definite answers to the standard question of infinitude of the solutions as well as several other surprising results.</p>\n<p><strong>Sums of squares, sums of cubes, and modern number theory</strong><br>\n<a href=\"https://kimballmartin.github.io/papers/quatcubforms.pdf\" target=\"_blank\">https://kimballmartin.github.io/papers/quatcubforms.pdf</a><br>\nOct 14, 2015 These are notes which grew out of a talk for general math graduate students with the aim of starting from the questions \\Which numbers are sums of two squares?\" and \\Which numbers are sums of two cubes?\" and going on a tour of many central topics in modern number theory. In the notes, I discuss composition laws, class groups, L-functions, modular forms, and elliptic curves, ending with the ...</p>\n<p><strong>Sum of Cubes and Beyond – Math Fun Facts</strong><br>\n<a href=\"https://math.hmc.edu/funfacts/sum-of-cubes-and-beyond/\" target=\"_blank\">https://math.hmc.edu/funfacts/sum-of-cubes-and-beyond/</a><br>\nOct 26, 2019 Hence the set of numbers {1,2,…,n} has the property that the sum of its cubes is the square of its sum. Are there any other collections of numbers with this property? Yes, and the following method is guaranteed to generate such a set. Pick a number, any number. Did I hear you say 63?Presentation Suggestions: If you are short on time, you can just present the sum of cubes fact. The Math Behind the Fact: From number theory, multiplicative functions are functions f defined over the positive integers that satisfy f(xy)=f(x)f(y) whenever integers x,y have no common factors.</p>\n<p><strong>Divisor function - Wikipedia</strong><br>\n<a href=\"https://en.wikipedia.org/wiki/Divisor_function\" target=\"_blank\">https://en.wikipedia.org/wiki/Divisor_function</a><br>\nDec 26, 2008 Sum of cubes of divisors, σ3(n) up to n = 250. In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer.Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum.</p>\n</details>\n<details><summary>Gathered key details from:<p></p>\n<ul>\n<li><a href=\"https://nntdm.net/papers/nntdm-19/NNTDM-19-1-01-13.pdf\" target=\"_blank\">NNTDM-19-1-01-13.pdf</a></li></ul></summary>\n\n<h3>Query</h3>\n<p>What is the mathematical proof of Proposition 1 about the divisor function τ(n) and CS-sets? Provide the complete proof and mathematical details.</p>\n<h3>Response</h3>\n<p>Proposition 1. For each natural n, denote by τ(n) the number of positive divisors of n. Then<br>\n⟨τ(d) :d|n,d ≥1⟩is a CS-set, where τ is taken over the positive divisors of n.<br>\nProof. The proposition is clearly true for prime powers since for each prime pand each natural<br>\nn, the corresponding set of pn−1 is<br>\n⟨τ(p0),τ(p1),...,τ (pn−1)⟩= ⟨1,2,...,n ⟩.<br>\nτ is well-known to be a multiplicative function [7], meaning that if a,b are relatively prime inte-<br>\ngers then τ(ab) =τ(a)τ(b). By elementary multiplicative number theory, the functions<br>\nf(n) =<br>\n∑<br>\nd|n<br>\nτ(d) and F(n) =<br>\n∑<br>\nd|n<br>\n[τ(d)]3<br>\nmust also be multiplicative. All we need now is thatF(n) = [f(n)]2. Since this is already true for<br>\nprime powers, the rest follows from the prime factorization of nand the multiplicative property<br>\nof f and F. □</p>\n</details>\n<details><summary>Response planned</summary><br>\n I found the exact theorem statement and proof. The theorem is indeed attributed to Liouville (1857) and states that for any positive integer n, if we consider all divisors of n and count the number of divisors of each divisor, then the sum of the cubes of these counts equals the square of their sum. I found both the historical reference and a modern mathematical proof using multiplicative functions.<p></p>\n<ol>\n<li>State the theorem precisely using mathematical notation</li>\n<li>Provide the example from the sources to illustrate</li>\n<li>Present the complete mathematical proof using multiplicative functions</li>\n<li>Explain the key mathematical concepts involved</li>\n</ol>\n</details><h2>Liouville's Divisor Sum Theorem</h2>\n<p><strong>Theorem Statement:</strong> For any positive integer <math display=\"inline\"><mi>n</mi></math>, let <math display=\"inline\"><mi>τ</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>d</mi><mo symmetric=\"false\" stretchy=\"false\">)</mo></math> denote the number of positive divisors of <math display=\"inline\"><mi>d</mi></math>. Then:</p>\n<p><math display=\"block\"><munder><mo movablelimits=\"false\">∑</mo><mrow><mi>d</mi><mi>|</mi><mi>n</mi></mrow></munder><mo symmetric=\"false\" stretchy=\"false\">[</mo><mi>τ</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>d</mi><mo symmetric=\"false\" stretchy=\"false\">)</mo><msup><mo symmetric=\"false\" stretchy=\"false\">]</mo><mn>3</mn></msup><mo>=</mo><msup><mrow><mo stretchy=\"true\">[</mo><munder><mo movablelimits=\"false\">∑</mo><mrow><mi>d</mi><mi>|</mi><mi>n</mi></mrow></munder><mi>τ</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>d</mi><mo symmetric=\"false\" stretchy=\"false\">)</mo><mo stretchy=\"true\">]</mo></mrow><mn>2</mn></msup></math></p>\n<p>where the sums are taken over all positive divisors <math display=\"inline\"><mi>d</mi></math> of <math display=\"inline\"><mi>n</mi></math>.</p>\n<p>This theorem was proven by Joseph Liouville in 1857<sup class=\"trigger\"><a href=\"https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/generalising-sums-of-cubes-equal-to-squares-of-sums/064A51DC4FAB6AEA9BCCC21A7ED62A70#:~:text=David%20Pagni,%5B4%2C%20p\" target=\"_blank\">1</a><span class=\"tooltip top center citation\"><a href=\"https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/generalising-sums-of-cubes-equal-to-squares-of-sums/064A51DC4FAB6AEA9BCCC21A7ED62A70#:~:text=David%20Pagni,%5B4%2C%20p\"><span class=\"source\"><img alt=\"\" src=\"/proxy/favicons?c=lQ30-cxV_E2Vdyqb6hrbuYgId7qu7hecjO7o1T7aqQDFc7XpcLttL1Ryala-GNHJi-PhJWT8zz50mE6K-yLSXc5-RKUQ_eQ4jOtvBLm666M%3D\"> www.cambridge.org</span><b>1. Generalising ‘Sums of cubes equal to squares of sums’</b></a><span class=\"passage\">David Pagni drew attention to a result which is ascribed by Dickson [2, p. 286] to Liouville (1857), that the sum of the cubes of the number of divisors of each of the divisors of an integer, is equal to the square of their sum. For example, the divisors of 6 are 1, 2, 3, and 6, which have 1, 2, 2, and 4 divisors respectively, and Pagni observed, as have others, including Mason et al. [4, p ...</span></span></sup>.</p>\n<h2>Example</h2>\n<p>Consider <math display=\"inline\"><mi>n</mi><mo>=</mo><mn>6</mn></math>. The divisors of 6 are: 1, 2, 3, and 6.</p>\n<ul>\n<li><math display=\"inline\"><mi>τ</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mn>1</mn><mo symmetric=\"false\" stretchy=\"false\">)</mo><mo>=</mo><mn>1</mn></math> (divisors: 1)</li>\n<li><math display=\"inline\"><mi>τ</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mn>2</mn><mo symmetric=\"false\" stretchy=\"false\">)</mo><mo>=</mo><mn>2</mn></math> (divisors: 1, 2)</li>\n<li><math display=\"inline\"><mi>τ</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mn>3</mn><mo symmetric=\"false\" stretchy=\"false\">)</mo><mo>=</mo><mn>2</mn></math> (divisors: 1, 3)</li>\n<li><math display=\"inline\"><mi>τ</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mn>6</mn><mo symmetric=\"false\" stretchy=\"false\">)</mo><mo>=</mo><mn>4</mn></math> (divisors: 1, 2, 3, 6)</li>\n</ul>\n<p>Left side: <math display=\"inline\"><msup><mn>1</mn><mn>3</mn></msup><mo>+</mo><msup><mn>2</mn><mn>3</mn></msup><mo>+</mo><msup><mn>2</mn><mn>3</mn></msup><mo>+</mo><msup><mn>4</mn><mn>3</mn></msup><mo>=</mo><mn>1</mn><mo>+</mo><mn>8</mn><mo>+</mo><mn>8</mn><mo>+</mo><mn>64</mn><mo>=</mo><mn>81</mn></math></p>\n<p>Right side: <math display=\"inline\"><mo symmetric=\"false\" stretchy=\"false\">(</mo><mn>1</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>4</mn><msup><mo symmetric=\"false\" stretchy=\"false\">)</mo><mn>2</mn></msup><mo>=</mo><msup><mn>9</mn><mn>2</mn></msup><mo>=</mo><mn>81</mn></math> ✓<sup class=\"trigger\"><a href=\"https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/generalising-sums-of-cubes-equal-to-squares-of-sums/064A51DC4FAB6AEA9BCCC21A7ED62A70#:~:text=Full%20document,more%20information.\" target=\"_blank\">1</a><span class=\"tooltip top center citation\"><a href=\"https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/generalising-sums-of-cubes-equal-to-squares-of-sums/064A51DC4FAB6AEA9BCCC21A7ED62A70#:~:text=Full%20document,more%20information.\"><span class=\"source\"><img alt=\"\" src=\"/proxy/favicons?c=lQ30-cxV_E2Vdyqb6hrbuYgId7qu7hecjO7o1T7aqQDFc7XpcLttL1Ryala-GNHJi-PhJWT8zz50mE6K-yLSXc5-RKUQ_eQ4jOtvBLm666M%3D\"> www.cambridge.org</span><b>1. Generalising ‘Sums of cubes equal to squares of sums’ | The Mathematical Gazette | Cambridge Core</b></a><span class=\"passage\">Full document cited. View source for more information.</span></span></sup></p>\n<h2>Mathematical Proof</h2>\n<p><strong>Proof:</strong> The proof relies on the multiplicative property of the divisor function <math display=\"inline\"><mi>τ</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>n</mi><mo symmetric=\"false\" stretchy=\"false\">)</mo></math>.</p>\n<p><strong>Step 1:</strong> For prime powers, the theorem holds trivially.<br>\nFor a prime <math display=\"inline\"><mi>p</mi></math> and natural number <math display=\"inline\"><mi>k</mi></math>, the divisors of <math display=\"inline\"><msup><mi>p</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup></math> are <math display=\"inline\"><msup><mi>p</mi><mn>0</mn></msup><mo>,</mo><msup><mi>p</mi><mn>1</mn></msup><mo>,</mo><mi>…</mi><mo>,</mo><msup><mi>p</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup></math>, which have <math display=\"inline\"><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi>…</mi><mo>,</mo><mi>k</mi></math> divisors respectively. The set <math display=\"inline\"><mo symmetric=\"false\" stretchy=\"false\">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi>…</mi><mo>,</mo><mi>k</mi><mo symmetric=\"false\" stretchy=\"false\">}</mo></math> satisfies the cube-sum property:<br>\n<math display=\"block\"><munderover><mo movablelimits=\"false\">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover><msup><mi>i</mi><mn>3</mn></msup><mo>=</mo><msup><mrow><mo stretchy=\"true\">[</mo><munderover><mo movablelimits=\"false\">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover><mi>i</mi><mo stretchy=\"true\">]</mo></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mo stretchy=\"true\">[</mo><mfrac><mrow><mi>k</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo symmetric=\"false\" stretchy=\"false\">)</mo></mrow><mrow><mn>2</mn></mrow></mfrac><mo stretchy=\"true\">]</mo></mrow><mn>2</mn></msup></math></p>\n<p><strong>Step 2:</strong> The divisor function <math display=\"inline\"><mi>τ</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>n</mi><mo symmetric=\"false\" stretchy=\"false\">)</mo></math> is multiplicative, meaning that for relatively prime integers <math display=\"inline\"><mi>a</mi></math> and <math display=\"inline\"><mi>b</mi></math>:<br>\n<math display=\"block\"><mi>τ</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>a</mi><mi>b</mi><mo symmetric=\"false\" stretchy=\"false\">)</mo><mo>=</mo><mi>τ</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>a</mi><mo symmetric=\"false\" stretchy=\"false\">)</mo><mi>τ</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>b</mi><mo symmetric=\"false\" stretchy=\"false\">)</mo></math></p>\n<p><strong>Step 3:</strong> Define the functions:</p>\n<ul>\n<li><math display=\"inline\"><mi>f</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>n</mi><mo symmetric=\"false\" stretchy=\"false\">)</mo><mo>=</mo><msub><mo movablelimits=\"false\">∑</mo><mrow><mi>d</mi><mi>|</mi><mi>n</mi></mrow></msub><mi>τ</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>d</mi><mo symmetric=\"false\" stretchy=\"false\">)</mo></math></li>\n<li><math display=\"inline\"><mi>F</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>n</mi><mo symmetric=\"false\" stretchy=\"false\">)</mo><mo>=</mo><msub><mo movablelimits=\"false\">∑</mo><mrow><mi>d</mi><mi>|</mi><mi>n</mi></mrow></msub><mo symmetric=\"false\" stretchy=\"false\">[</mo><mi>τ</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>d</mi><mo symmetric=\"false\" stretchy=\"false\">)</mo><msup><mo symmetric=\"false\" stretchy=\"false\">]</mo><mn>3</mn></msup></math></li>\n</ul>\n<p>Since <math display=\"inline\"><mi>τ</mi></math> is multiplicative, both <math display=\"inline\"><mi>f</mi></math> and <math display=\"inline\"><mi>F</mi></math> are also multiplicative functions<sup class=\"trigger\"><a href=\"https://nntdm.net/papers/nntdm-19/NNTDM-19-1-01-13.pdf#:~:text=Proposition%201.,is%20%E2%9F%A8%CF%84%28p0%29%2C%CF%84%28p1%29%2C&amp;text=%2C%CF%84%20%28pn%E2%88%921%29%E2%9F%A9%3D,%28pn%E2%88%921%29%E2%9F%A9%3D%20%E2%9F%A81%2C2%2C&amp;text=%2Cn%20%E2%9F%A9.,F.%20%E2%96%A1\" target=\"_blank\">2</a><span class=\"tooltip top center citation\"><a href=\"https://nntdm.net/papers/nntdm-19/NNTDM-19-1-01-13.pdf#:~:text=Proposition%201.,is%20%E2%9F%A8%CF%84%28p0%29%2C%CF%84%28p1%29%2C&amp;text=%2C%CF%84%20%28pn%E2%88%921%29%E2%9F%A9%3D,%28pn%E2%88%921%29%E2%9F%A9%3D%20%E2%9F%A81%2C2%2C&amp;text=%2Cn%20%E2%9F%A9.,F.%20%E2%96%A1\"><span class=\"source\"><img alt=\"\" src=\"/proxy/favicons?c=lQ30-cxV_E2Vdyqb6hrbuYgId7qu7hecjO7o1T7aqQDFc7XpcLttL1Ryala-GNHJitYHHsRpaLIpphKouGJcCugTGdi7kcEDStdjbGg766o%3D\"> nntdm.net</span><b>2. NNTDM-19-1-01-13.pdf</b></a><span class=\"passage\">Proposition 1. For each natural n, denote by τ(n) the number of positive divisors of n. Then ⟨τ(d) :d|n,d ≥1⟩is a CS-set, where τ is taken over the positive divisors of n. Proof. The proposition is clearly true for prime powers since for each prime pand each natural n, the corresponding set of pn−1 is ⟨τ(p0),τ(p1),...,τ (pn−1)⟩= ⟨1,2,...,n ⟩. τ is well-known to be a multiplicative function [7], meaning that if a,b are relatively prime inte- gers then τ(ab) =τ(a)τ(b). By elementary ...</span></span></sup>.</p>\n<p><strong>Step 4:</strong> We need to show that <math display=\"inline\"><mi>F</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>n</mi><mo symmetric=\"false\" stretchy=\"false\">)</mo><mo>=</mo><mo symmetric=\"false\" stretchy=\"false\">[</mo><mi>f</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>n</mi><mo symmetric=\"false\" stretchy=\"false\">)</mo><msup><mo symmetric=\"false\" stretchy=\"false\">]</mo><mn>2</mn></msup></math> for all <math display=\"inline\"><mi>n</mi></math>.</p>\n<p>Since this equality holds for prime powers (Step 1), and both <math display=\"inline\"><mi>F</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>n</mi><mo symmetric=\"false\" stretchy=\"false\">)</mo></math> and <math display=\"inline\"><mo symmetric=\"false\" stretchy=\"false\">[</mo><mi>f</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>n</mi><mo symmetric=\"false\" stretchy=\"false\">)</mo><msup><mo symmetric=\"false\" stretchy=\"false\">]</mo><mn>2</mn></msup></math> are multiplicative functions, the equality extends to all positive integers by the fundamental theorem of arithmetic.</p>\n<p>Specifically, if <math display=\"inline\"><mi>n</mi><mo>=</mo><msubsup><mi>p</mi><mn>1</mn><mrow><msub><mi>a</mi><mn>1</mn></msub></mrow></msubsup><msubsup><mi>p</mi><mn>2</mn><mrow><msub><mi>a</mi><mn>2</mn></msub></mrow></msubsup><mi>⋯</mi><msubsup><mi>p</mi><mi>k</mi><mrow><msub><mi>a</mi><mi>k</mi></msub></mrow></msubsup></math>, then:<br>\n<math display=\"block\"><mi>F</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>n</mi><mo symmetric=\"false\" stretchy=\"false\">)</mo><mo>=</mo><mi>F</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><msubsup><mi>p</mi><mn>1</mn><mrow><msub><mi>a</mi><mn>1</mn></msub></mrow></msubsup><mo symmetric=\"false\" stretchy=\"false\">)</mo><mo>⋅</mo><mi>F</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><msubsup><mi>p</mi><mn>2</mn><mrow><msub><mi>a</mi><mn>2</mn></msub></mrow></msubsup><mo symmetric=\"false\" stretchy=\"false\">)</mo><mi>⋯</mi><mi>F</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><msubsup><mi>p</mi><mi>k</mi><mrow><msub><mi>a</mi><mi>k</mi></msub></mrow></msubsup><mo symmetric=\"false\" stretchy=\"false\">)</mo></math><br>\n<math display=\"block\"><mo symmetric=\"false\" stretchy=\"false\">[</mo><mi>f</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>n</mi><mo symmetric=\"false\" stretchy=\"false\">)</mo><msup><mo symmetric=\"false\" stretchy=\"false\">]</mo><mn>2</mn></msup><mo>=</mo><mo symmetric=\"false\" stretchy=\"false\">[</mo><mi>f</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><msubsup><mi>p</mi><mn>1</mn><mrow><msub><mi>a</mi><mn>1</mn></msub></mrow></msubsup><mo symmetric=\"false\" stretchy=\"false\">)</mo><msup><mo symmetric=\"false\" stretchy=\"false\">]</mo><mn>2</mn></msup><mo>⋅</mo><mo symmetric=\"false\" stretchy=\"false\">[</mo><mi>f</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><msubsup><mi>p</mi><mn>2</mn><mrow><msub><mi>a</mi><mn>2</mn></msub></mrow></msubsup><mo symmetric=\"false\" stretchy=\"false\">)</mo><msup><mo symmetric=\"false\" stretchy=\"false\">]</mo><mn>2</mn></msup><mi>⋯</mi><mo symmetric=\"false\" stretchy=\"false\">[</mo><mi>f</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><msubsup><mi>p</mi><mi>k</mi><mrow><msub><mi>a</mi><mi>k</mi></msub></mrow></msubsup><mo symmetric=\"false\" stretchy=\"false\">)</mo><msup><mo symmetric=\"false\" stretchy=\"false\">]</mo><mn>2</mn></msup></math></p>\n<p>Since <math display=\"inline\"><mi>F</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><msubsup><mi>p</mi><mi>i</mi><mrow><msub><mi>a</mi><mi>i</mi></msub></mrow></msubsup><mo symmetric=\"false\" stretchy=\"false\">)</mo><mo>=</mo><mo symmetric=\"false\" stretchy=\"false\">[</mo><mi>f</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><msubsup><mi>p</mi><mi>i</mi><mrow><msub><mi>a</mi><mi>i</mi></msub></mrow></msubsup><mo symmetric=\"false\" stretchy=\"false\">)</mo><msup><mo symmetric=\"false\" stretchy=\"false\">]</mo><mn>2</mn></msup></math> for each prime power, we have <math display=\"inline\"><mi>F</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>n</mi><mo symmetric=\"false\" stretchy=\"false\">)</mo><mo>=</mo><mo symmetric=\"false\" stretchy=\"false\">[</mo><mi>f</mi><mo symmetric=\"false\" stretchy=\"false\">(</mo><mi>n</mi><mo symmetric=\"false\" stretchy=\"false\">)</mo><msup><mo symmetric=\"false\" stretchy=\"false\">]</mo><mn>2</mn></msup></math> for all <math display=\"inline\"><mi>n</mi></math><sup class=\"trigger\"><a href=\"https://nntdm.net/papers/nntdm-19/NNTDM-19-1-01-13.pdf#:~:text=Proposition%201.,is%20%E2%9F%A8%CF%84%28p0%29%2C%CF%84%28p1%29%2C&amp;text=%2C%CF%84%20%28pn%E2%88%921%29%E2%9F%A9%3D,%28pn%E2%88%921%29%E2%9F%A9%3D%20%E2%9F%A81%2C2%2C&amp;text=%2Cn%20%E2%9F%A9.,F.%20%E2%96%A1\" target=\"_blank\">2</a><span class=\"tooltip top center citation\"><a href=\"https://nntdm.net/papers/nntdm-19/NNTDM-19-1-01-13.pdf#:~:text=Proposition%201.,is%20%E2%9F%A8%CF%84%28p0%29%2C%CF%84%28p1%29%2C&amp;text=%2C%CF%84%20%28pn%E2%88%921%29%E2%9F%A9%3D,%28pn%E2%88%921%29%E2%9F%A9%3D%20%E2%9F%A81%2C2%2C&amp;text=%2Cn%20%E2%9F%A9.,F.%20%E2%96%A1\"><span class=\"source\"><img alt=\"\" src=\"/proxy/favicons?c=lQ30-cxV_E2Vdyqb6hrbuYgId7qu7hecjO7o1T7aqQDFc7XpcLttL1Ryala-GNHJitYHHsRpaLIpphKouGJcCugTGdi7kcEDStdjbGg766o%3D\"> nntdm.net</span><b>2. NNTDM-19-1-01-13.pdf</b></a><span class=\"passage\">Proposition 1. For each natural n, denote by τ(n) the number of positive divisors of n. Then ⟨τ(d) :d|n,d ≥1⟩is a CS-set, where τ is taken over the positive divisors of n. Proof. The proposition is clearly true for prime powers since for each prime pand each natural n, the corresponding set of pn−1 is ⟨τ(p0),τ(p1),...,τ (pn−1)⟩= ⟨1,2,...,n ⟩. τ is well-known to be a multiplicative function [7], meaning that if a,b are relatively prime inte- gers then τ(ab) =τ(a)τ(b). By elementary ...</span></span></sup>.</p>\n<h2>Key Mathematical Insight</h2>\n<p>The theorem works because the divisor counts form sequences that are element-by-element products of consecutive integer sequences <math display=\"inline\"><mo symmetric=\"false\" stretchy=\"false\">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi>…</mi><mo>,</mo><mi>t</mi><mo symmetric=\"false\" stretchy=\"false\">}</mo></math>, which have the fundamental property that their sum of cubes equals the square of their sum<sup class=\"trigger\"><a href=\"https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/generalising-sums-of-cubes-equal-to-squares-of-sums/064A51DC4FAB6AEA9BCCC21A7ED62A70#:~:text=Full%20document,more%20information.\" target=\"_blank\">1</a><span class=\"tooltip top center citation\"><a href=\"https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/generalising-sums-of-cubes-equal-to-squares-of-sums/064A51DC4FAB6AEA9BCCC21A7ED62A70#:~:text=Full%20document,more%20information.\"><span class=\"source\"><img alt=\"\" src=\"/proxy/favicons?c=lQ30-cxV_E2Vdyqb6hrbuYgId7qu7hecjO7o1T7aqQDFc7XpcLttL1Ryala-GNHJi-PhJWT8zz50mE6K-yLSXc5-RKUQ_eQ4jOtvBLm666M%3D\"> www.cambridge.org</span><b>1. Generalising ‘Sums of cubes equal to squares of sums’ | The Mathematical Gazette | Cambridge Core</b></a><span class=\"passage\">Full document cited. View source for more information.</span></span></sup>. The multiplicative structure of the divisor function preserves this property across all positive integers.</p>","md":"## Liouville's Divisor Sum Theorem\n\n**Theorem Statement:** For any positive integer $n$, let $\\tau(d)$ denote the number of positive divisors of $d$. Then:\n\n$$\\sum_{d|n} [\\tau(d)]^3 = \\left[\\sum_{d|n} \\tau(d)\\right]^2$$\n\nwhere the sums are taken over all positive divisors $d$ of $n$.\n\nThis theorem was proven by Joseph Liouville in 1857[^1].\n\n## Example\n\nConsider $n = 6$. The divisors of 6 are: 1, 2, 3, and 6.\n- $\\tau(1) = 1$ (divisors: 1)\n- $\\tau(2) = 2$ (divisors: 1, 2) \n- $\\tau(3) = 2$ (divisors: 1, 3)\n- $\\tau(6) = 4$ (divisors: 1, 2, 3, 6)\n\nLeft side: $1^3 + 2^3 + 2^3 + 4^3 = 1 + 8 + 8 + 64 = 81$\n\nRight side: $(1 + 2 + 2 + 4)^2 = 9^2 = 81$ ✓[^1]\n\n## Mathematical Proof\n\n**Proof:** The proof relies on the multiplicative property of the divisor function $\\tau(n)$.\n\n**Step 1:** For prime powers, the theorem holds trivially.\nFor a prime $p$ and natural number $k$, the divisors of $p^{k-1}$ are $p^0, p^1, \\ldots, p^{k-1}$, which have $1, 2, \\ldots, k$ divisors respectively. The set $\\{1, 2, \\ldots, k\\}$ satisfies the cube-sum property:\n$$\\sum_{i=1}^k i^3 = \\left[\\sum_{i=1}^k i\\right]^2 = \\left[\\frac{k(k+1)}{2}\\right]^2$$\n\n**Step 2:** The divisor function $\\tau(n)$ is multiplicative, meaning that for relatively prime integers $a$ and $b$:\n$$\\tau(ab) = \\tau(a)\\tau(b)$$\n\n**Step 3:** Define the functions:\n- $f(n) = \\sum_{d|n} \\tau(d)$\n- $F(n) = \\sum_{d|n} [\\tau(d)]^3$\n\nSince $\\tau$ is multiplicative, both $f$ and $F$ are also multiplicative functions[^2].\n\n**Step 4:** We need to show that $F(n) = [f(n)]^2$ for all $n$.\n\nSince this equality holds for prime powers (Step 1), and both $F(n)$ and $[f(n)]^2$ are multiplicative functions, the equality extends to all positive integers by the fundamental theorem of arithmetic.\n\nSpecifically, if $n = p_1^{a_1} p_2^{a_2} \\cdots p_k^{a_k}$, then:\n$$F(n) = F(p_1^{a_1}) \\cdot F(p_2^{a_2}) \\cdots F(p_k^{a_k})$$\n$$[f(n)]^2 = [f(p_1^{a_1})]^2 \\cdot [f(p_2^{a_2})]^2 \\cdots [f(p_k^{a_k})]^2$$\n\nSince $F(p_i^{a_i}) = [f(p_i^{a_i})]^2$ for each prime power, we have $F(n) = [f(n)]^2$ for all $n$[^2].\n\n## Key Mathematical Insight\n\nThe theorem works because the divisor counts form sequences that are element-by-element products of consecutive integer sequences $\\{1, 2, \\ldots, t\\}$, which have the fundamental property that their sum of cubes equals the square of their sum[^1]. The multiplicative structure of the divisor function preserves this property across all positive integers.","profile":{"id":null,"name":null,"subtitle":"State of the art deep research (>30 secs)","model":"ki_research","model_name":"Research","model_provider":"kagi","model_input_limit":1000000,"recommended":false,"model_info":"\n\n<div class=\"heading\">\n <div class=\"left\">\n \n \n \n <svg class=\"model-icon icon-xs\" data-model=\"ki_research\">\n <use href=\"#icon-kagi\"/>\n </svg>\n <h3>\n \n <span class=\"model-provider\">Kagi</span>\n \n <span class=\"model-name\">none</span>\n </h3>\n </div>\n\n \n <a href=\"/settings?p=custom_assistant&id=none\">\n <svg class=\"icon-sm edit-icon\" aria-hidden=\"true\">\n <use href=\"#icon-pencil\" />\n </svg>\n </a>\n \n</div>\n\n\n<hr />\n<ul>\n \n <li>\n <h4>\n Cost\n </h4>\n <span>\n \n \n \n\n \n \n \n\n <div class=\"score\" aria-label=\"Expensive\">\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-cost --high\"></div>\n \n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-cost --high\"></div>\n \n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-cost --high\"></div>\n \n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-cost --high\"></div>\n \n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-cost --high\"></div>\n \n \n </div>\n</span>\n </li>\n <li>\n <h4>\n Quality\n </h4>\n <span>\n \n \n \n\n \n \n \n\n <div class=\"score\" aria-label=\"Rated 5 out of 5\">\n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n </div>\n</span>\n </li>\n <li>\n <h4>\n Speed\n </h4>\n <span>\n \n \n \n\n \n \n \n\n <div class=\"score\" aria-label=\"Rated 1 out of 5\">\n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n </div>\n</span>\n </li>\n <li>\n <h4>\n Privacy\n </h4>\n <span>\n \n \n \n\n \n \n \n\n <div class=\"score\" aria-label=\"Rated 4 out of 5\">\n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"filled-bubble\"></div>\n \n \n \n \n <div aria-hidden=\"true\" class=\"unfilled-bubble\"></div>\n \n \n </div>\n</span>\n </li>\n <li class=\"divider\" aria-hidden=\"true\"></li>\n <li>\n <h4>Last updated</h4>\n <span class=\"date\">\n \n \n 2025-11-18\n \n \n </span>\n </li>\n \n</ul>\n","model_provider_name":"Kagi","internet_access":true,"personalizations":true,"shortcut":null,"is_default_profile":false},"metadata":"<li>\n <span class=\"attribute\">Model</span>\n <span class=\"value\"><i class=\"icon-xs mr-2\"><svg><use href=\"#icon-kagi\"></use></svg></i>Research</span>\n</li>\n<li>\n <span class=\"attribute\">Version</span>\n <span class=\"value\">ki_research-2025-11-18</span>\n</li>\n<li>\n <span class=\"attribute\">Speed (tok/s)</span>\n <span class=\"value\">64</span>\n</li>\n<li>\n <span class=\"attribute\">Tokens</span>\n <span class=\"value\">15,232</span>\n</li>\n<li>\n <span class=\"attribute\">Cost / Total ($)</span>\n <span class=\"value\">0.14 / 0.14</span>\n</li>\n<li>\n <span class=\"attribute\">End to end time (s)</span>\n <span class=\"value\">52.57</span>\n</li>\n\n","documents":[]}]