Loading...
Please wait, while we are loading the content...
Similar Documents
On sets of three consecutive integers which are quadratic residues of primes
| Content Provider | Semantic Scholar |
|---|---|
| Author | Bennett, Albert Arnold |
| Copyright Year | 1925 |
| Abstract | In this paper we shall prove the following theorems. THEOREM I. For each prime, p, for which there are as many as three incongruent squares, there is a set of three consecutive residues (admitting zero and negative numbers as residues) which are squares, modulo p. THEOREM II. For p = 11, and for each prime p greater than 17, (and for no other primes), there is a set of thrçe consecutive least positive (non-zero) residues which are squares, modulo p. The problemt of finding three consecutive integers which are quadratic residues of a prime, p, is equivalent to the formally more general problem of finding two quantities, x, y, (y^O), such that x, y, x-\-y, x — y, are proportional to squares in the domain, $ since we then have (x/y) — 1, x/y, (xly)-\-l as consecutive squares in the domain. We may show that for residues with respect to a modulus the condition is equivalent to the existence of a square of the form§ u v (u-\-v) (u—v). By taking u = x, v = y, we see that the condition is necessary. |
| Starting Page | 411 |
| Ending Page | 412 |
| Page Count | 2 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/S0002-9904-1925-04078-1 |
| Alternate Webpage(s) | http://www.ams.org/journals/bull/1925-31-08/S0002-9904-1925-04078-1/S0002-9904-1925-04078-1.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/S0002-9904-1925-04078-1 |
| Volume Number | 31 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
