### Abstract

Erdõs and Straus conjectured that for any positive integer n ≥ 2 the equation 4/n = 1/x + 1/y + 1/x has a solution in positive integers z, y, and z. Let m > k ≥ 3 and E_{m,k}(N) =| {n ≤ N | m/n = 1/t_{1} + ⋯ + 1/t_{k}has no solution with t_{i} ∈ ℕ} | . We show that parametric solutions can be used to find upper bounds on E_{m,k}(N) where the number of parameters increases exponentially with k. This enables us to prove E_{m,k}(N) ≪ N exp(-c_{n,k}(log N)^{1-1/2k-1-1}) with c_{m,k} > 0. This improves upon earlier work by Viola (1973) and Shen (1986), and is an "exponential generalization" of the work of Vaughan (1970), who considered the case k = 3.

Original language | English |
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Pages (from-to) | 3209-3227 |

Number of pages | 19 |

Journal | Transactions of the American Mathematical Society |

Volume | 353 |

Issue number | 8 |

Publication status | Published - 2001 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Transactions of the American Mathematical Society*,

*353*(8), 3209-3227.

**Sums of k unit fractions.** / Elsholtz, Christian.

Research output: Contribution to journal › Article › Research › peer-review

*Transactions of the American Mathematical Society*, vol. 353, no. 8, pp. 3209-3227.

}

TY - JOUR

T1 - Sums of k unit fractions

AU - Elsholtz, Christian

PY - 2001

Y1 - 2001

N2 - Erdõs and Straus conjectured that for any positive integer n ≥ 2 the equation 4/n = 1/x + 1/y + 1/x has a solution in positive integers z, y, and z. Let m > k ≥ 3 and Em,k(N) =| {n ≤ N | m/n = 1/t1 + ⋯ + 1/tkhas no solution with ti ∈ ℕ} | . We show that parametric solutions can be used to find upper bounds on Em,k(N) where the number of parameters increases exponentially with k. This enables us to prove Em,k(N) ≪ N exp(-cn,k(log N)1-1/2k-1-1) with cm,k > 0. This improves upon earlier work by Viola (1973) and Shen (1986), and is an "exponential generalization" of the work of Vaughan (1970), who considered the case k = 3.

AB - Erdõs and Straus conjectured that for any positive integer n ≥ 2 the equation 4/n = 1/x + 1/y + 1/x has a solution in positive integers z, y, and z. Let m > k ≥ 3 and Em,k(N) =| {n ≤ N | m/n = 1/t1 + ⋯ + 1/tkhas no solution with ti ∈ ℕ} | . We show that parametric solutions can be used to find upper bounds on Em,k(N) where the number of parameters increases exponentially with k. This enables us to prove Em,k(N) ≪ N exp(-cn,k(log N)1-1/2k-1-1) with cm,k > 0. This improves upon earlier work by Viola (1973) and Shen (1986), and is an "exponential generalization" of the work of Vaughan (1970), who considered the case k = 3.

UR - http://www.scopus.com/inward/record.url?scp=33646936573&partnerID=8YFLogxK

M3 - Article

VL - 353

SP - 3209

EP - 3227

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 8

ER -