A Pellian Equation with Primes and Applications to $$D(-1)$$ D ( - 1 ) -Quadruples

D(−1)-quadruples and Products of Two Primes

A D(−1)-quadruple is a set of positive integers {a, b, c, d}, with a < b < c < d, such that the product of any two elements from this set is of the form 1+n2 for some integer n. Dujella and Fuchs showed that any such D(−1)-quadruple satisfies a = 1. The D(−1) conjecture states that there is no D(−1)-quadruple. If b = 1+ r2, c = 1+ s2 and d = 1+ t2, then it is known that r, s, t, b, c and d are ...

The number of D(−1)-quadruples

In this paper, we first show that for any fixed D(−1)-triple {1, b, c} with b < c, there exist at most two d’s such that {1, b, c, d} is a D(−1)-quadruple with c < d. Using this result, we further show that there exist at most 10 D(−1)-quadruples. AMS subject classifications: 11D09, 11D45

On Diophantine quintuples and D(−1)-quadruples

In this paper the known upper bound 10 for the number of Diophantine quintuples is reduced to 6.8·10. The key ingredient for the improvement is that certain individual bounds on parameters are now combined with a more efficient counting of tuples, and estimated by sums over divisor functions. As a side effect, we also improve the known upper bound 4 ·10 for the number of D(−1)-quadruples to 5 ·...

NOTES ON PRIMES P ⌘ 1 mod D AND A P � 1 / D ⌘ 1 mod

Let d > 0 be a squarefree integer and a be an integer, which is not 1 nor a square. Let P(a,d)(x) be the number of primes p  x such that p ⌘ 1 mod d and a(p 1)/d ⌘ 1 mod p. Numerical data indicate that the function as approximately equal to a constant multiple of ⇡(x)/(d'(d)) for su ciently large x, where ⇡(x) is the number of primes up to x and '(d) is the Euler-' function. The involved const...

The 1-D Wave Equation