[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81898-en":3,"doc-seo-81898-105":31,"detail-sidebar-cat-0-en-105":93},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},81898,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","A Sharp Lower Bound for Some Reciprocal Rado Numbers","Let fr(k) denote the smallest n such that every r-coloring of {1,2,...,n} yields a monochromatic solution to a reciprocal Rado-type equation in variables x1,...,xk that need not be distinct. This paper establishes improved lower bounds: fr(2) ≥ 4r/2 for all r ≥ 1 and fr(k) ≥ (2r−1)kr for all k ≥ 3 and r ≥ 1. For r=2, it characterizes cases where f2(k)=3k^2 (including infinitely many k) and gives a bound f2(k)≥3k^2+1 when k is an odd prime power, alongside new computations for f2(k) and f3(k) and a coefficient generalization.","A sharp lower bound for some reciprocal Rado numbers  \nCollier Gaiser∗ Mojtaba Ramezanpour†  \narXiv :2607 .04373v1 [math .CO] 5 Jul 2026  \nAbstract  \nLet fr (k) be the smallest n such that every r-coloring of {1, 2 ,..., n} has a monochromatic solution to the equation  \nwhere x 1 , x2 , . . . , xk are not necessarily distinct. In this paper, we prove that fr (2) ≥ 4r /2 for all r ≥ 1, and fr (k) ≥ (2r −1)kr for all k ≥ 3 and r ≥ 1. When r = 2, we show that, if k = 3·2m for some positive integer m, then f2 (k) = 3k2 ; and if k = pm for some odd prime number p and positive integer m, then f2 (k) ≥ 3k2 + 1 . We also provide new computational results for f2 (k) and f3 (k), as well as a generalization of our lower bounds for f2 (k) to equations with general coefficients.  \nKeywords: arithmetic Ramsey theory, unit fractions, Rado numbers  \nAMS 2020 subject classification: 05D10; 11B75  \n1 Introduction  \nIn arithmetic Ramsey theory, by the celebrated Rado’s theorem [14] (see also [10, Chapter 9]), for all integers k ≥ 2 and r ≥ 1, if n is large enough, then every r-coloring of {1, 2 ,..., n} has a monochromatic solution to the equation  \nx 1 + x2 + ··· + xk = xk+1 . (1)  \nLet Rr(k) be the smallest positive integer n such that every r-coloring of {1, 2 ,..., n} has a monochromatic solution to Equation (1), where x 1 , x2 , . . . , xk are not necessarily distinct. Due to Schur’s theorem [15] (see also [10, Chapter 8]), the values of Rr(2) are known as Schur numbers and it is known that R 1 (2) = 2, R2 (2) = 5, R3 (2) = 14, and R4 (2) = 45 . Only very recently in 2018, using a computer-generated proof that required two petabytes of space, Heule [7] determined that R5 (2) = 161 . All other Schur numbers are still undetermined. We refer interested readers to [1] for recent developments on Schur numbers.  \nSchur [15] (see also [10, Theorem 8.9]) proved that Rr(2) ≥ (3r − 1)/2 + 1 for all r ≥ 1. Zn´am [17] generalized this result and proved that, for all k ≥ 2 and r ≥ 1,  Rr(k) ≥ k~~ ~~−k~~ ~~1 [(k + 1) r − 1] + 1 .  \n∗ Department of Mathematics, Community College of Aurora, Aurora, CO 80011, United States of America. Email: [colliergaiser@gmail.com](colliergaiser@gmail.com)  \n†Community College of Aurora, Aurora, CO 80011, United States of America. Email: [mrramezanpour7698@gmail.com](mrramezanpour7698@gmail.com)  \nWe note that Zn´am [17] proved the above lower bound in 1966, and then Beutelspacher and Brestovansky [2] reproved this lower bound in 1982 . When r = 2 and 3, the exact values for Rr(k) are known for all k and they coincide with the lower bound of Zn´am: Beutelspacher and Brestovansky [2] proved that R2 (k) = k 2 + k − 1 for all k ≥ 2, and, recently in 2019, Boza, Mar´ın, Revuelta, and Sanz [3] confirmed that R3 (k) = k3 + 2k2 − 2 for all k ≥ 2.  \nBrown and R¨odl [4], and Lefmann [11] independently proved that, among other things, Rado’s theorem still holds when the variables are replaced with their reciprocals. This implies that, for all positive integers k ≥ 2 and r ≥ 1, if n is large enough, then every r-coloring of {1, 2 ,..., n} has a monochromatic solution to the equation  \n 1   1   1   1   \n+ + ··· + = (2)  \nx 1 x2 xk xk+1 .  \nLet fr(k) be the smallest positive integer n such that every r-coloring of {1, 2 ,..., n} has a monochromatic solution to Equation (2), where x 1 , x2 , . . . , xk are not necessarily distinct. Brown and R¨odl [4] proved that f2 (k) ≤ k2 (k2 − k + 1)(k2 + k − 1) and the first author [5] improved this upper bound to f2 (k) ≤ 6k(k + 1)(k + 2) in 2024 . Using a result of Boza, Mar´ın, Revuelta, and Sanz [3], the first author [5] also showed a polynomial upper bound for f3 (k) .  \nAs for the lower bound, Tejaswi and Thangdurai [16] proved that fr(2) ≥ 3 · 2r+3 for all r ≥ 3, and the first author [5] observed that fr(k) ≥ kr for all k ≥ 2 and r ≥ 1. In this paper, we first improve both lower bounds.  \nTheorem 1.1 . For all r ≥ 1, we have  \nfr(2) ≥ 4r /2;  \nand, for all k ≥ 3 and r ≥ 1, we have  \nfr (k) ","cbCaiezR7jaHlmuJ","https://ap.wps.com/l/cbCaiezR7jaHlmuJ","pdf",362293,6,1,15,"English","en",105,"# Introduction\n## Background on Rado numbers and Schur numbers\n## Reciprocal Rado numbers and definitions of fr(k)\n## Statement of main lower bound theorems\n## Sharpness, explicit cases for r=2, and conjecture\n## Extensions to equations with general coefficients","[{\"question\":\"What does fr(k) represent in this paper?\",\"answer\":\"fr(k) is the smallest n such that any r-coloring of {1,2,...,n} guarantees a monochromatic solution to the reciprocal Rado-type equation, with x1,...,xk not necessarily distinct.\"},{\"question\":\"What lower bounds are proved for fr(2) and fr(k) in general?\",\"answer\":\"The paper proves fr(2) ≥ 4r/2 for all r ≥ 1, and for all k ≥ 3 and r ≥ 1 it proves fr(k) ≥ (2r−1)kr.\"},{\"question\":\"How is f2(k) determined for some families of k?\",\"answer\":\"For r=2, the results show f2(k)=3k^2 when k=3·2^m, while for odd prime powers k it proves f2(k) ≥ 3k^2+1, and computational evidence supports a broader characterization via an odd-prime-power condition.\"}]","A Sharp Lower Bound for Some Reciprocal Rado Numbers | 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does fr(k) represent in this paper?","Question",{"text":77,"@type":78},"fr(k) is the smallest n such that any r-coloring of {1,2,...,n} guarantees a monochromatic solution to the reciprocal Rado-type equation, with x1,...,xk not necessarily distinct.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"What lower bounds are proved for fr(2) and fr(k) in general?",{"text":82,"@type":78},"The paper proves fr(2) ≥ 4r/2 for all r ≥ 1, and for all k ≥ 3 and r ≥ 1 it proves fr(k) ≥ (2r−1)kr.",{"name":84,"@type":75,"acceptedAnswer":85},"How is f2(k) determined for some families of k?",{"text":86,"@type":78},"For r=2, the results show f2(k)=3k^2 when k=3·2^m, while for odd prime powers k it proves f2(k) ≥ 3k^2+1, and computational evidence supports a broader characterization via an odd-prime-power 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