Solutions In Integers X1 Research Articles

A conjecture on integer arithmetic which implies that there is an algorithm which to each Diophantine equation assigns an integer which is greater than the heights of integer (non-negative integer, rational) solutions, if these solutions form a finite set

We conjecture that if a system S ⊆ {xi + xj = xk, xi · xj = xk : i, j, k ∈ {1, . . . , n}} has only finitely many solutions in integers x1, . . . , xn, then each such solution (x1, . . . , xn) satisfies max (|x1|, . . . , |xn|) ≤ 22 . The conjecture implies that there is an algorithm which to each Diophantine equation assigns an integer which is greater than the heights of integer (non-negative integer, rational) solutions, if these solutions form a finite set. We describe an algorithm whose execution never terminates. If the conjecture is true, then the algorithm sequentially displays all integers n ≥ 2. If the conjecture is false, then the algorithm has a finite output which ends with an integer tuple (a1, . . . , an), where n ≥ 4, 22 n−1 < max (|a1|, . . . , |an|) ≤ 22 , and the system {xi + xj = xk : (i, j, k ∈ {1, . . . , n}) ∧ (ai + aj = ak)} ∪ {xi · xj = xk : (i, j, k ∈ {1, . . . , n}) ∧ (ai · aj = ak)} is a counterexample to the conjecture. The algorithm is implemented in MuPAD and Pascal. Mathematics Subject Classification: 03B30, 11U05

On a conjecture of Watson

Let Qn be a real indefinite quadratic form in n variables x1, x2,…, xn, of determinant D ≠ 0 and of type (r, s), 0 &lt; r &lt; n, n = r + s. Let σ denote the signature of Qn so that σ = r − s. It is known (see e.g. Blaney(4)) that, given any real numbers c1 c2, …, cn, there exists a constant C depending upon n and σ only such that the inequalityhas a solution in integers x1, x2, …, xn. Let Cr, s denote the infimum of all such constants. Clearly Cr, s = Cs, r, so we need consider non-negative signatures only. For n = 2, C1, 1 = ¼ follows from a classical result of Minkowski on the product of two linear forms. When n = 3, Davenport (5) proved that C2, 1 = 27/100. For all n and σ = 0, Birch (3) proved that Cr, r = ¼. In 1962, Watson(18) determined the values of Cr, s for all n ≥ 21 and for all signatures σ. He proved thatWatson also conjectured that (1·2) holds for all n ≥ 4. Dumir(6) proved Watson's conjecture for n = 4. For n = 5, it was proved by Hans-Gill and Madhu Raka(7, 8). The author (12) has proved the conjecture for σ = 1 and all n. In the preceding paper (13) we proved that C5, 1 = 1. In this paper we prove Watson's conjecture for σ = 2, 3 and 4.

Pairs of additive equations III: quintic equations

We consider R simultaneous equations of additive typewhere the coefficients aij are integers. Artin's conjecture, for additive forms, is that the equations (1) have a non-trivial solution in integers x1,…,xN provided that they have a non-trivial real solution, which is clearly satisfied when k is odd, and

The inhomogeneous minimum of quadratic forms of signature ± 1

Let Qr be a real indefinite quadratic form in r variables of determinant D ≠ 0 and of type (r1, r2), 0 &lt; r1 &lt; r, r = r1 + r2, S = r1 − r2 being the signature of Qr. It is known (e.g. Blaney (3)) that, given any real numbers c1, c2,…, cr, there exists a constant C depending only on r and s such that the inequalityhas a solution in integers x1, x2, …, xr.

Cubic forms in sixteen variables

It is proved that if C ( x 1 , ..., x n ) is any cubic form in n variables, with integral coefficients, then the equation C ( x 1 , ..., x n ) = 0 has a solution in integers x 1 , ..., x n not all 0, provided n is at least 16. This is an improvement upon earlier results (Davenport 1959, 1962).

Cubic forms in 29 variables

It is proved that if C (x 1 ..., x n) is any cubic form in n variables, with integral coefficients, then the equation C ( x 1 x n ) = 0 has a solution in integers x 1 ..., x n not all 0, provided n is at least 29. This is an improvement on a previous result (Davenport 1959).