We give an explicit formula for the Frobenius number of triples associated with the Diophantine equation $x^2+y^2=z^3$, that is, the largest positive integer that can only be represented in p ways by combining the three integers of the solutions of $x^2+y^2=z^3$. For the equation $x^2+y^2=z^2$, the Frobenius number has already been given. Our approach can be extended to the general equation $x^2+y^2=z^r$ for $r>3$.

Ishitsuka et al. [‘Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic’, Int. J. Number Theory 16(4) (2020), 881–905] found all points on the Fermat quartic ${F_4\colon x^4+y^4=z^4}$ over quadratic extensions of ${\mathbb {Q}}(\zeta _8)$, where $\zeta _8$ is the eighth primitive root of unity $e^{i\pi /4}$. Using Mordell’s technique, we give an alternative proof for the result of Ishitsuka et al. and extend it to the rational function field ${\mathbb {Q}}({\zeta _8})(T_1,T_2,\ldots ,T_n)$.

Given any polynomial in two variables of degree at most three with rational integer coefficients, we obtain a new search bound to decide effectively if it has a zero with rational integer coefficients. On the way we encounter a natural problem of estimating singular points. We solve it using elementary invariant theory but an optimal solution would seem to be far from easy even using the full power of the standard Height Machine.

In this paper we investigate the quantity of diagonal quartic surfaces $a_0 X_0^4 + a_1 X_1^4 + a_2 X_2^4 +a_3 X_3^4 = 0$ which have a Brauer–Manin obstruction to the Hasse principle. We are able to find an asymptotic formula for the quantity of such surfaces ordered by height. The proof uses a generalization of a method of Heath-Brown on sums over linked variables. We also show that there exists no uniform formula for a generic generator in this family.

We generalise two quartic surfaces studied by Swinnerton-Dyer to give two infinite families of diagonal quartic surfaces which violate the Hasse principle. Standard calculations of Brauer–Manin obstructions are exhibited.

In the 1993 Western Number Theory Conference, Richard Guy proposed Problem 93:31, which asks for integers n representable by ${(x+y+z)^3}/{xyz}$ , where $x,\,y,\,z$ are integers, preferably with positive integer solutions. We show that the representation $n={(x+y+z)^3}/{xyz}$ is impossible in positive integers $x,\,y,\,z$ if $n=4^{k}(a^2+b^2)$ , where $k,\,a,\,b\in \mathbb {Z}^{+}$ are such that $k\geq 3$ and $2\nmid a+b$ .

For any fixed nonzero integer h, we show that a positive proportion of integral binary quartic forms F do locally everywhere represent h, but do not globally represent h. We order classes of integral binary quartic forms by the two generators of their ring of ${\rm GL}_{2}({\mathbb Z})$ -invariants, classically denoted by I and J.

Aigner showed in 1934 that nontrivial quadratic solutions to $x^4 + y^4 = 1$ exist only in $\mathbb Q(\sqrt {-7})$ . Following a method of Mordell, we show that nontrivial quadratic solutions to $x^4 + 2^ny^4 = 1$ arise from integer solutions to the equations $X^4 \pm 2^nY^4 = Z^2$ investigated in 1853 by V. A. Lebesgue.

Assuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant $C$ such that for any elliptic curve $E/\mathbb{Q}$ and non-torsion point $P\,\in \,E\left( \mathbb{Q} \right)$, there is at most one integral multiple $\left[ n \right]P$ such that $n\,>\,C$. The proof is a modification of a proof of Ingram giving an unconditional, but not uniform, bound. The new ingredient is a collection of explicit formulæ for the sequence $v\left( {{\Psi }_{n}} \right)$ of valuations of the division polynomials. For $P$ of non-singular reduction, such sequences are already well described in most cases, but for $P$ of singular reduction, we are led to define a new class of sequences called elliptic troublemaker sequences, which measure the failure of the Néron local height to be quadratic. As a corollary in the spirit of a conjecture of Lang and Hall, we obtain a uniform upper bound on $\widehat{h}\left( P \right)/h\left( E \right)$ for integer points having two large integral multiples.

We solve the Diophantine equation $Y^{2}=X^{3}+k$ for all nonzero integers $k$ with $|k|\leqslant 10^{7}$. Our approach uses a classical connection between these equations and cubic Thue equations. The latter can be treated algorithmically via lower bounds for linear forms in logarithms in conjunction with lattice-basis reduction.

Frey and Jarden asked if any abelian variety over a number field $K$ has the infinite Mordell–Weil rank over the maximal abelian extension ${{K}^{\text{ab}}}$. In this paper, we give an affirmative answer to their conjecture for the Jacobian variety of any smooth projective curve $C$ over $K$ such that $\sharp C\left( {{K}^{\text{ab}}} \right)\,=\,\infty $ and for any abelian variety of $\text{G}{{\text{L}}_{2}}$-type with trivial character.

A number is squareful if the exponent of every prime in its prime factorization is at least two. In this paper, we give, for a fixed $l$, the number of pairs of squareful numbers $n$, $n+l$ such that $n$is less than a given quantity.

It is proven that if k ≥ 2 is an integer and d is a positive integer such that the product of any two distinct elements of the set increased by 1 is a perfect square, then d = 4k or d = 64k 5−48k 3+8k. Together with a recent result of Fujita, this shows that all Diophantine quadruples of the form {k − 1, k + 1, c, d} are regular.

J.W.S. Cassels gave a solution to the problem of determining all instances of the sum of three consecutive cubes being a square. This amounts to finding all integer solutions to the Diophantine equation $y^2=3x(x^2+2)$. We describe an alternative approach to solving not only this equation, but any equation of the type $y^2=nx(x^2+2)$, with $n$ a natural number. Moreover, we provide an explicit upper bound for the number of solutions of such Diophantine equations. The method we present uses the ingenious work of Wilhelm Ljunggren, and a recent improvement by the authors.

In 1913 Frobenius conjectured that for any positive integer m, there exists at most one pair of integers (x, y) with 0 ≤ x ≤ y ≤ m such that (x, y, m) is a solution to the Markoff equation: x2 + y2 + m2 = 3xym. We show this is true if either m, 3m — 2 or 3m + 2 is prime, twice a prime or four times a prime.

It is shown that the equation xyz = x+y+z has unit solutions in only four not totally real cubic fields: two fields which are real and two fields which are imaginary. These fields are then listed.