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Waring problem with the Ramanujan $\tau$-function

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M. Z. Garaev, V. C. Garcia, S. V. Konyagin

Number Theory · Submitted 6 Jul 2006

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove that for any integer $N$ the diophantine equation $$ \sum_{i=1}^{74000}\tau(n_i)=N $$ has a solution in positive integers $n_1, n_2,..., n_{74000}$ satisfying the condition $$ \max_{1\le i\le 74000}n_i\ll |N|^{2/11}+1.

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arXiv IDmath/0607169 (version 1)
Pith IDV242EQCV, Pith's permanent record of this paper and its review
CitationM. Z. Garaev, S. V. Konyagin, V. C. Garcia. "Waring problem with the Ramanujan $\tau$-function." arXiv:math/0607169 (2006). https://pith.science/paper/V242EQCV
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,040 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.14 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.05 · mentions code or data availability 0.01.

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arXiv listing abstract v1. Results and the main claim are underlined; .

Let $\tau(n)$ be the Ramanujan $\tau$-function. We prove that for any integer $N$ the diophantine equation $$ \sum_{i=1}^{74000}\tau(n_i)=N $$ has a solution in positive integers $n_1, n_2,..., n_{74000}$ satisfying the condition $$ \max_{1\le i\le 74000}n_i\ll |N|^{2/11}+1. $$ We also consider similar questions in the residue ring modulo a large prime $p.$

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Let $\tau(n)$ be the Ramanujan $\tau$-function. background 1.00 0.07 0.01 asserted 0.73
s1 We prove that for any integer $N$ the diophantine equation $$ \sum_{i=1}^{74000}\tau(n_i)=N $$ has a solution in positive integers $n_1, n_2,..., n_{74000}$ satisfying the condition $$ \max_{1\le i\le 74000}n_i\ll |N|^{2/11}+1. result 0.97 0.96 0.94 asserted 1.00
s2 $$ We also consider similar questions in the residue ring modulo a large prime $p.$ contribution 0.91 0.84 0.03 asserted 0.55

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Let $\tau(n)$ be the Ramanujan $\tau$-function. We prove that for any integer $N$ the diophantine equation $$ \sum_{i=1}^{74000}\tau(n_i)=N $$ has a solution in positive integers $n_1, n_2,..., n_{74000}$ satisfying the condition $$ \max_{1\le i\le 74000}n_i\ll |N|^{2/11}+1. $$ We also consider similar questions in the residue ring modulo a large prime $p.$

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