On products and shifted products of residues modulo \(p\). (English) Zbl 1168.11009
It is shown that if \(p\) is prime and \(\mathcal A, \mathcal B, \mathcal C, \mathcal D\) are large subsets of \(\mathbb Z_p\), then the equation \(ab+1=cd, a\in \mathcal A, b\in \mathcal B, c\in \mathcal C, d\in \mathcal D\) has a solution. This is a multiplicative analogue of the author’s earlier result where it was shown that the equation \(a+b=cd\) has a solution. Several interesting known and new results are shown to follow as a consequence, such as the solvability of both Legendre symbol equations \((\frac{a+b}{p})=1\) and \((\frac{a'+b'}{p})=-1\), where \(a'\in \mathcal A, b'\in \mathcal B\). The proof uses simple combinatorial arguments and summation techniques.
Reviewer: Anitha Srinivasan (Mumbai)
MSC:
11D45 | Counting solutions of Diophantine equations |
11B75 | Other combinatorial number theory |
11L40 | Estimates on character sums |
11A07 | Congruences; primitive roots; residue systems |