I don't know what you mean by "recursive" in this case, but you might be interested in the solution that I posted here for generating all rational solutions using one guessed solution. The idea is that the slope of a line through the guessed point is rational if and only if the other point at which it intersects the curve is a rational point.
If you are looking for integers solutions, you can find all rational solutions and then look into when the coordinates are integers, say by using divisibility theorems. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Recursive equations for quadratic diophantine equations. Ask Question.
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A linear Diophantine equation equates the sum of two or more monomialseach of degree 1 in one of the variables, to a constant. An exponential Diophantine equation is one in which exponents on terms can be unknowns. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations.
In more technical language, they define an algebraic curvealgebraic surfaceor more general object, and ask about the lattice points on it. The word Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandriawho made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra.
The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis. While individual equations present a kind of puzzle and have been considered throughout history, the formulation of general theories of Diophantine equations beyond the theory of quadratic forms was an achievement of the twentieth century.
In the following Diophantine equations, wxyand z are the unknowns and the other letters are given constants:. The solutions are described by the following theorem:. Thus, if the equation has a solution, then c must be a multiple of d. More generally, every system of linear Diophantine equations may be solved by computing the Smith normal form of its matrix, in a way that is similar to the use of the reduced row echelon form to solve a system of linear equations over a field.
Using matrix notation every system of linear Diophantine equations may be written. The system to be solved may thus be rewritten as. If this condition is fulfilled, the solutions of the given system are. Hermite normal form may also be used for solving systems of linear Diophantine equations.
However, Hermite normal form does not directly provide the solutions; to get the solutions from the Hermite normal form, one has to successively solve several linear equations. Nevertheless, Richard Zippel wrote that the Smith normal form "is somewhat more than is actually needed to solve linear diophantine equations. Instead of reducing the equation to diagonal form, we only need to make it triangular, which is called the Hermite normal form.
The Hermite normal form is substantially easier to compute than the Smith normal form. Integer linear programming amounts to finding some integer solutions optimal in some sense of linear systems that include also inequations. Thus systems of linear Diophantine equations are basic in this context, and textbooks on integer programming usually have a treatment of systems of linear Diophantine equations.A general quadratic Diophantine equation in two variables and is given by.
The slightly more general second-order equation. In particular, all solutions of. For quadratic Diophantine equations in more than two variables, there exist additional deep results due to C. Pell equations, as well as the analogous equation with a minus sign on the right, can be solved by finding the continued fraction for. The more complicated equation. However, if a single solution to 5 is known, other solutions can be found using the standard technique for the Pell equation.
The following table summarizes possible representation of primes of given forms, where and are positive integers. No odd primes other than those indicated share these properties Nagellp. As a part of the study of Waring's problemit is known that every positive integer is a sum of no more than 4 positive squares ; Lagrange's four-square theoremthat every "sufficiently large" integer is a sum of no more than 4 positive squaresand that every integer is a sum of at most 3 signed squares.
If zero is counted as a square, both positive and negative numbers are included, and the order of the two squares is distinguished, Jacobi showed that the number of ways a number can be written as the sum of two squares the function is four times the excess of the number of divisors of the form over the number of divisors of the form.
Given an initial solution to the equation. Piezas, pers. Call a Diophantine equation consisting of finding a sum of th powers which is equal to a sum of th powers a " equation.
To solve the equation, note that every prime of the form can be expressed as the sum of two relatively prime squares in exactly one way. A set of integers satisfying the 2. The number of solutions are given by the sum of squares function.
Quadratic diophantine equations
Another similar identity is the Euler four-square identity. Degen's eight-square identity holds for eight squares, but no other number, as proved by Cayley. The two-square identity underlies much of trigonometrythe four-square identity some of quaternionsand the eight-square identity, the Cayley algebra a noncommutative nonassociative algebra; Bell In an unpublished proof, Euler showed that the quadratic Diophantine equation.
Rather amazingly, these can be given analytically by. The first few solutions for2, 3, Beiler, A. New York: Dover, pp. Bell, E. The Development of Mathematics, 2nd ed. New York: McGraw-Hill, p.Springer Book Archives: eBooks only 8. Authors: AndreescuTitu, AndricaDorin. This monograph treats the classical theory of quadratic Diophantine equations and guides the reader through the last two decades of computational techniques and progress in the area.
These new techniques combined with the latest increases in computational power shed new light on important open problems. The authors motivate the study of quadratic Diophantine equations with excellent examples, open problems and applications. Moreover, the exposition aptly demonstrates many applications of results and techniques from the study of Pell-type equations to other problems in number theory. The book is intended for advanced undergraduate and graduate students as well as researchers.
It challenges the reader to apply not only specific techniques and strategies, but also to employ methods and tools from other areas of mathematics, such as algebra and analysis. It is well written, well organized, and contains a wealth of material that one does not expect to find in a book of its size, with full proofs of scores of theorems.
Feldman, Choice, Vol. Voutier, Mathematical Reviews, March, Only valid for books with an ebook version. Springer Reference Works are not included.
Mathematics Number Theory and Discrete Mathematics. Developments in Mathematics Free Preview. Includes both theoretical and computational examples Explores new computational techniques for quadratic diophantine equations Techniques presented will shed light on important open problems see more benefits. Buy eBook. Buy Hardcover. Buy Softcover.Number theory or arithmetic or higher arithmetic in older usage is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions.
German mathematician Carl Friedrich Gauss — said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics. Integers can be considered either in themselves or as solutions to equations Diophantine geometry. Questions in number theory are often best understood through the study of analytical objects for example, the Riemann zeta function that encode properties of the integers, primes or other number-theoretic objects in some fashion analytic number theory.
One may also study real numbers in relation to rational numbers, for example, as approximated by the latter Diophantine approximation. The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory".
The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. The earliest historical find of an arithmetical nature is a fragment of a table: the broken clay tablet Plimpton Larsa, Mesopotamiaca.
The triples are too many and too large to have been obtained by brute force. The heading over the first column reads: "The takiltum of the diagonal which has been subtracted such that the width The table's layout suggests  that it was constructed by means of what amounts, in modern language, to the identity. It is not known what these applications may have been, or whether there could have been any; Babylonian astronomyfor example, truly came into its own only later.
It has been suggested instead that the table was a source of numerical examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, Babylonian algebra in the secondary-school sense of "algebra" was exceptionally well developed. Much earlier sources  state that Thales and Pythagoras traveled and studied in Egypt.
The Pythagorean tradition spoke also of so-called polygonal or figurate numbers. We know of no clearly arithmetical material in ancient Egyptian or Vedic sources, though there is some algebra in both. The Chinese remainder theorem appears as an exercise  in Sunzi Suanjing 3rd, 4th or 5th century CE. There is also some numerical mysticism in Chinese mathematics, [note 5] but, unlike that of the Pythagoreans, it seems to have led nowhere.
I mean some kind of parametrization which gives all solutions but no points which do not satisfy the equation. It is conjectured also by Sierpinski that there are infinitely many primes of the above form but this still remains open. So, your question cannot be answered yet. Let me just add that for solving quadratic diophantine equations in 2 variables, i. Optionally, the program shows all steps of the solution, similar as a human might do when solving the equation by hand.
The website also describes the algorithm used. There are infinitely many integer solutions, so you need to specify what you mean by "can one find all". If you mean "Are all solutions given by a finite set of easily computed parametrized families? This sign ambiguity nicely cancels the invariance under sign change in Euler's parametrization. Carnahan Sep 24 '13 at showed parametrization by cosets of congruence subgroups of SL 2,Z. What is missing is parametrization of SL 2,Z and its congruence subgroups which can be found in L.
Polynomial parametrization for the solutions of Diophantine equations and arithmetic groupsAnnals of Math.
Zbl This algorithm works by the Hasse principle. Otherwise, we analyse the signature of the quadratic form.
Otherwise, there are infinitely many solutions as intersecting one solution with an appropriately-chosen plane yields a generalised Pell equation.
Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Is there an algorithm to solve quadratic Diophantine equations? Ask Question. Asked 6 years, 10 months ago. Active 3 years, 10 months ago. Viewed 14k times. I'm looking for all triplets. The idea you gave was already given in stackexchange. But it seems to be hard to find all solutions. Active Oldest Votes. Konstantinos Gaitanas Konstantinos Gaitanas 3, 1 1 gold badge 14 14 silver badges 32 32 bronze badges.
Gerry Myerson Gerry Myerson Mar 31 '15 at Sorry I do not have reference to Adleman paper or Garey Johnson. Apr 2 '15 at Read the papers. Stefan Kohl Stefan Kohl 17k 9 9 gold badges 59 59 silver badges bronze badges. HTMan it gives me the same recursive formula as you got, but not the starting value e. Are you sure you put negative signs in the right places?A Diophantine equation is an equation relating integer or sometimes natural number or whole number quanitites.
Finding the solution or solutions to a Diophantine equation is closely tied to modular arithmetic and number theory. Often, when a Diophantine equation has infinitely many solutions, parametric form is used to express the relation between the variables of the equation.
A Diophantine equation in the form is known as a linear combination. If two relatively prime integers and are written in this form withthe equation will have an infinite number of solutions. More generally, there will always be an infinite number of solutions when. Ifthen there are no solutions to the equation. To see why, consider the equation.
However, will never be a multiple ofhence, no solutions exist. Now consider the case where. If and are relatively prime, then all solutions are obviously in the form for all integers. If they are not, we simply divide them by their greatest common divisor.
A Pythagorean triple is a set of three integers that satisfy the Pythagorean Theorem. There are three main methods of finding Pythagorean triples:. If is an odd number, then is a Pythagorean triple. Ifis a Pythagorean triple. For anywe have is a Pythagorean triple. A equation of form has no integer solutions, as follows: We assume that the equation does have integer solutions, and consider the solution which minimizes. Let this solution be.
If then their GCD must satsify. The solution would then be a solution less thanwhich contradicts our assumption. Thus, this equation has no integer solutions. Ifwe then proceed with casework, in. Note that every square, and therefore every fourth power, is either or. The proof of this is fairly simple, and you can show it yourself.
Case This would implya contradiction. This would implya contradiction since we assumed. Case 3:and. We also know that squares are either or.
Diophantine Equation--2nd Powers
Thus, all fourth powers are either or. This is a contradiction, as implies is odd, and implies is even. QED [Oops, this doesn't work.