Add to ORKGProblems found in Diophantus's books on Arithmetics prompted intense research activity in recent years. The modern treatment evolved from the unifying notion of D(n)-m-sets. After a short overview of the main questions in this area; we concentrate on D(-1)-sets. It is conjectured that no D(-1)-quadrupleb exist, but the issue is still open. In the literature one finds several properties a hypothetical D(-1)-quadruple necessarily has. After sketching the strategy producing such results, a novel approach is pointed out. This survey is based on work in progress performed jointly with N. C. Bonciocat (Bucharest, Romania) and M. Mignote (Strasbourg, France)
AbstractIn this paper, we consider the D(−1)-triple {1,k2+1,(k+1)2+1}. We extend the result obtained...
We solve the problem of Diophantus for integers of the quadratic field Q(√−3) by finding a D(z)-quad...
This note gives a summary of the paper [M. Cipu, Y. Fujita, M. Mignotte, Two-parameter families of u...
Quadruples (a; b; c; d) of positive integers a < b < c < d with the property that the product of any...
For a nonzero integer n, a set of distinct nonzero integers {a_{1} : a_{2} : a_{m}} such that a_{i}a...
In this paper, we first show that for any fixed D(-1)-triple {1,b,c} with b<c, there exist at most t...
Let $z$ be an element of a commutative ring $R$. A Diophantine quadruple with the property $D(z)$, o...
summary:Let $(a_1,\dots , a_m)$ be an $m$-tuple of positive, pairwise distinct integers. If for all ...
A D(-1)-quadruple is a set of positive integers {a, b, c, d}, with a r. A particular instance yields...
summary:Let $(a_1,\dots , a_m)$ be an $m$-tuple of positive, pairwise distinct integers. If for all ...
In this paper we prove that every Diophantine quadruple in ℝ [X] is regular. In other words, we prov...
A D(-1)-quadruple is a set of positive integers {a, b, c, d}, with a r. A particular instance yields...
We solve the problem of Diophantus for integers of the quadratic field Q(√−3) by finding a D(z)-quad...
We solve the problem of Diophantus for integers of the quadratic field Q(√−3) by finding a D(z)-quad...
In this paper we prove that there does not exist a set of four non-zero polynomials from (mathbb{Z}[...
AbstractIn this paper, we consider the D(−1)-triple {1,k2+1,(k+1)2+1}. We extend the result obtained...
We solve the problem of Diophantus for integers of the quadratic field Q(√−3) by finding a D(z)-quad...
This note gives a summary of the paper [M. Cipu, Y. Fujita, M. Mignotte, Two-parameter families of u...
Quadruples (a; b; c; d) of positive integers a < b < c < d with the property that the product of any...
For a nonzero integer n, a set of distinct nonzero integers {a_{1} : a_{2} : a_{m}} such that a_{i}a...
In this paper, we first show that for any fixed D(-1)-triple {1,b,c} with b<c, there exist at most t...
Let $z$ be an element of a commutative ring $R$. A Diophantine quadruple with the property $D(z)$, o...
summary:Let $(a_1,\dots , a_m)$ be an $m$-tuple of positive, pairwise distinct integers. If for all ...
A D(-1)-quadruple is a set of positive integers {a, b, c, d}, with a r. A particular instance yields...
summary:Let $(a_1,\dots , a_m)$ be an $m$-tuple of positive, pairwise distinct integers. If for all ...
In this paper we prove that every Diophantine quadruple in ℝ [X] is regular. In other words, we prov...
A D(-1)-quadruple is a set of positive integers {a, b, c, d}, with a r. A particular instance yields...
We solve the problem of Diophantus for integers of the quadratic field Q(√−3) by finding a D(z)-quad...
We solve the problem of Diophantus for integers of the quadratic field Q(√−3) by finding a D(z)-quad...
In this paper we prove that there does not exist a set of four non-zero polynomials from (mathbb{Z}[...
AbstractIn this paper, we consider the D(−1)-triple {1,k2+1,(k+1)2+1}. We extend the result obtained...
We solve the problem of Diophantus for integers of the quadratic field Q(√−3) by finding a D(z)-quad...
This note gives a summary of the paper [M. Cipu, Y. Fujita, M. Mignotte, Two-parameter families of u...