### Abstract

A set {b_{1},b_{2},...,b_{i}} ⊂ {1,2,...,N} is said to be a difference intersector set if {a_{1},a_{2},...,a_{s}} ⊂ {1,2,...,N}, j > ε{lunate}N imply the solvability of the equation a_{x} - a_{y} = b′; the notion of sum intersector set is defined similarly. The authors prove two general theorems saying that if a set {b_{1},b_{2},...,b_{i}} is well distributed simultaneously among and within all residue classes of small moduli then it must be both difference and sum intersector set. They apply these theorems to investigate the solvability of the equations (a_{x} - a_{y} p = + 1, (a_{u} - a_{v} p) = - 1, (a_{r} + a_{s} p) = + 1, (a_{t} + a_{z} p) = - 1 (where ( a p) denotes the Legendre symbol) and to show that "almost all" sets form both difference and sum intersector sets.

Original language | English |
---|---|

Pages (from-to) | 430-450 |

Number of pages | 21 |

Journal | Journal of Number Theory |

Volume | 10 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1978 |

### Fingerprint

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**On differences and sums of integers, I.** / Erdős, P.; Sárközy, A.

Research output: Contribution to journal › Article

*Journal of Number Theory*, vol. 10, no. 4, pp. 430-450. https://doi.org/10.1016/0022-314X(78)90017-3

}

TY - JOUR

T1 - On differences and sums of integers, I

AU - Erdős, P.

AU - Sárközy, A.

PY - 1978

Y1 - 1978

N2 - A set {b1,b2,...,bi} ⊂ {1,2,...,N} is said to be a difference intersector set if {a1,a2,...,as} ⊂ {1,2,...,N}, j > ε{lunate}N imply the solvability of the equation ax - ay = b′; the notion of sum intersector set is defined similarly. The authors prove two general theorems saying that if a set {b1,b2,...,bi} is well distributed simultaneously among and within all residue classes of small moduli then it must be both difference and sum intersector set. They apply these theorems to investigate the solvability of the equations (ax - ay p = + 1, (au - av p) = - 1, (ar + as p) = + 1, (at + az p) = - 1 (where ( a p) denotes the Legendre symbol) and to show that "almost all" sets form both difference and sum intersector sets.

AB - A set {b1,b2,...,bi} ⊂ {1,2,...,N} is said to be a difference intersector set if {a1,a2,...,as} ⊂ {1,2,...,N}, j > ε{lunate}N imply the solvability of the equation ax - ay = b′; the notion of sum intersector set is defined similarly. The authors prove two general theorems saying that if a set {b1,b2,...,bi} is well distributed simultaneously among and within all residue classes of small moduli then it must be both difference and sum intersector set. They apply these theorems to investigate the solvability of the equations (ax - ay p = + 1, (au - av p) = - 1, (ar + as p) = + 1, (at + az p) = - 1 (where ( a p) denotes the Legendre symbol) and to show that "almost all" sets form both difference and sum intersector sets.

UR - http://www.scopus.com/inward/record.url?scp=28844449017&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=28844449017&partnerID=8YFLogxK

U2 - 10.1016/0022-314X(78)90017-3

DO - 10.1016/0022-314X(78)90017-3

M3 - Article

AN - SCOPUS:28844449017

VL - 10

SP - 430

EP - 450

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 4

ER -