#### Glasnik Matematicki, Vol. 49, No. 2 (2014), 275-285.

### ON THE PRIME DIVISORS OF ELEMENTS OF A *D(-1)* QUADRUPLE

### Anitha Srinivasan

Department of Mathematics, Saint Louis University-Madrid campus, Avenida del Valle 34, 28003 Madrid, Spain

*e-mail:* `rsrinivasan.anitha@gmail.com`

**Abstract.**
In [6] it was shown that
if *{1, b, c, d}* is a *D(-1)* quadruple with *b < c < d * and *b=1+r*^{2}, then
*r* and *b* are not of the form *r=p*^{k},
*r=2p*^{k}, *b=p* or * b=2p*^{k}, where
*p* is an odd prime and *k* is a positive integer.
We show that an identical result holds for *c=1+s*^{2},
that is, the cases *s=p*^{k}, s=2p^{k}, c=p and * c=2p*^{k}
do not occur for the *D(-1)*
quadruple given above.
For the integer *d=1+x*^{2}, we show
that *d* is not prime and that *x*
is divisible by at least two distinct odd primes.
Furthermore, we present several infinite families of integers
*b*, such that the
*D(-1)* pair *{1, b}* cannot
be extended to a *D(-1)* quadruple.
For instance, we show that
if *r=5p* where *p* is an odd prime, then the
*D(-1)* pair *{1, r*^{2}+1} cannot
be extended to a *D(-1)* quadruple.

**2010 Mathematics Subject Classification.**
11D09, 11R29, 11E16.

**Key words and phrases.** Diophantine *m* tuples, binary quadratic forms, quadratic Diophantine equation.

**Full text (PDF)** (access from subscribing institutions only)
DOI: 10.3336/gm.49.2.03

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