Introduction
The study of number theory is a very old subject in mathematics, the first example being a broken clay tablet found in Mesopotamia that contained a list of Pythagorean triples. Because of the increase in computing power in recent years and advances in the field of quantum computing number theory has seen somewhat of a resurgence. Numbers that in the past took weeks to compute can now be obtained in a matter of minutes or even seconds. Even tough great advances have been made there are still unsolved problems in this field. For my Bachelor’s Thesis I have decided to focus on Diophantine equations.
The study of Diophantine equations dates back 1600 BC, and the first important problem was that of determining Pythagorean triples, which refers to finding solutions to the following equation: a^2 + b^2 = c^2.
Diophantine Equations are named after the Greek mathematician Diophantus, who was born between AD 201 and AD 215 and wrote a series of books called Arithmetica. Out of the 13 books only 6 survived. The method for solving the problems he proposed has become known as Diophantine Analysis. The first person to give the general solution for the linear equation ax + by = c was the Indian mathematician Brahmagupta.
Diophantus made contributions in mathematics that have had impact through the millenniums, and even in the present day. His most influential work is in numbers theory and also the methods he used for solving the problems. There are three major scholars who used and built upon his work, Vieta, Poincare and Fermat.
Another important name in the history of number theory and Diophantine Equations was Pierre de Fermat. One of Fermat’s legacies was the introduction of the infinite descent method which can solve a lot of difficult problem especially in the field of Diophantine equations. The method of infinite descent assumes a solution that is as small as possible, and the by some means we find another solution that is even smaller, thus contradicting the first assumption. This method can be used to prove negative results such as the equation x^4+ y^4= z^2 [1] having no nontrivial integer solutions, and also proving positive results.
A Diophantine equation is a polynomial equation where only integer solutions are sought out. The simplest Diophantine equation is in the form: ax + by = c, where a, b and c are given integers, x and y are the solutions, also integers. Solving this linear Diophantine equation requires following an ordered pattern of steps, which include using the Euclidean algorithm.
In the year 1900 David Hilbert posed a list of ten mathematical problems. The tenth problem on the list asked if there is a general algorithm that, for any given Diophantine equation can determine whether the equation has a solution with all the unknowns being integers. The answer to the question is negative as proved by Yuri Matiyasevich in 1970.
There are two general approaches to solving Diophantine equations: analytical and empirical.
The analytical approach requires finding a general solution that can find all the solutions to the equation. For example, if we have the equation ax + by = n where (x^*,y^*) is an integer solution, then all the integer solutions are of form (x^*+m b/gcⅆ(a,b) ,y^*-m a/gcⅆ(a,b) ) for some integer m.
The empirical approach requires finding an algorithm that determines solutions to the Diophantine equation on a finite domain.
For my Bachelor’s Thesis I will be focusing on using the empirical approach.
In the first chapter of my Thesis I will star by defining what a Diophantine equation is, presenting some of the famous results obtained by mathematicians and the different types of Diophantine equations that exist.
In the second chapter, firstly I will show how to analytically solve a linear Diophantine equation, providing an example on solving. On the second part of the chapter I will show the program used to partially solve Diophantine equations.
Diophantine m-tuples
A set of positive integers {a_1,a_2,…,a_m} is said to have the property D(n), where n is an integer, if a_i*a_j+n is a perfect square for all 1 ≤ i < j ≤ m. A set of this form is called a Diophantine m-tuple.
The first Diophantine quadruple was found by Fermat with the property D(1), and it was {1,3,8,120}.
An almost Diophantine quintuple we understand the set {a, b, c, d, e} which verifies 9 out of the 10 Diophantine conditions.
A quintuple with the property D(n) is an almost Diophantine quintuple that verifies all 10 of the Diophantine conditions.
The 10 Diophantine conditions are the following:
b ∙a+n= q_1^2,
c ∙a+n= q_2^2,
d ∙a+n= q_3^2,
e∙a+n= q_4^2,
c ∙b+n= q_5^2,
d ∙b+n= q_6^2,
e ∙b+n= q_7^2,
d ∙c+n= q_8^2,
e ∙c+n= q_9^2,
e ∙d+n= q_10^2,
Where q_1,q_2,…,q_10 are positive integers.
Chapter 1
A Diophantine equation is an equation of the form
f(x_(1 ),x_2 , . . . ,x_n) = 0,
(1.1)
where f is a given function and the unknowns x_1 ,x_2 , . . . ,x_n are required to be rational numbers or to be integers.
In the study of Diophantine Analysis there are a few questions that arose naturally:
Can we solve the equation?
Are there any solutions beyond the obvious ones?
Is the number of solutions finite or infinite?
Can we find all of the solutions in theory?
Can we find all of the solutions practically (having a list of solutions)?
We know about Diophantine equations that there are infinitely many pairs {a, b}, and also infinitely many Diophantine triples and quadruples. Also, a Diophantine pair can be extended to a Diophantine triple by adding a+b+2 ∙r to the set, where a ∙b+1= r^2. The same can be said about Diophantine triples {a, b, c}, we can extend them to quadruples if we have a ∙b+1= r^2, b ∙c+1= s^2, c ∙a+1= t^2, where r, s and t are positive integers. Then we have the Diophantine quadruple set {a, b, c, d} with d=a+b+c+2∙a∙b∙c+2∙r∙s∙t [2]. These types of quadruples are called regular.
We also know that there are no Diophantine 6-tuples, 7-tuples, 8-tuples etc. This was proven by Andrej Dujella in 2004.
There are different types of Diophantine equations, some of which have been studied more and have generated interest from mathematicians. One type of Diophantine equation is a linear Diophantine equation, for example ax+by=1 .
We have the following well-studied Diophantine equation
x^2+ny^2=1 (2.1)
where n is a given positive nonsquare integer, and we search for integer solutions for x and y. This is known as Pell’s equation and is named after the English mathematician John Pell. The equation was studied intensively in India and Brahmagupta developed a method for solving the equation in 628, about a thousand years before Pell.
We also have the following Diophantine equation which is quite interesting:
w^3+x^3=y^3+Z^3 (3.1)
The smallest nontrivial answer to this solution in integers is 〖12〗^3+1^3=9^3+〖10〗^3=1729 and it was given to G.H. Hardy by the Indian mathematician Ramanujan, in the year 1917. An interesting anecdote is that while Ramanujan was in hospital, he was visited by Hardy who had taken a cab with the number 1729.
A Diophantine equation that has additional variables in the form of exponents are called exponential Diophantine equations. Examples of these types of equations are the Ramanujan-Nagell equation: 2^n-7=x^2 and Beal’s conjecture: a^n+b^n=c^k. We do not have a general equation to solve such equations, but there have been attempts to solve particular cases. Most of these equations are solved by using ad-hoc methods such as trial and error.
Another type of equation is an infinite Diophantine equation. An example of this type of equation can be the following:
n=a^2+2b^2+3c^2+4d^2+.…, (4.1)
In layman’s term this can be interpreted as “In how many ways can a given integer n best written as the sum of a squared plus two times b squared plus three times c squared and so on”. The equation always has a solution if n is positive.
Probably the most famous out of all the Diophantine equations is Fermat’s Last Theorem.
Let us consider the following equation:
x^n+y^n=z^n (5.1)
where x, y and z are positive integers and n is an integer greater than 2. Fermat’s theorem says that the equation doesn’t have any integer solutions when n is greater than 2, except a trivial solution when one of the variables is 0.
The theorem was first proposed by the French mathematician Pierre de Fermat in 1637, when he wrote on the margin of a copy of Arithmetica that he had proof of this, but not enough space on the book to scribble it.
The first person to successfully prove this was Andrew Wiles in 1995, after more than 350. The theorem was even part of the Guinness Book of World Records as the most difficult problem in mathematics, mainly because there have been so many unsuccessful attempts to prove it.
Diophantine triples
In mathematics, a Diophantine m-tuple is a set of m positive integers {a_1,a_2,…,a_m} such that a_i*a_j+n is a perfect square for any 1 ≤ i < j ≤ m. A set of m positive rational numbers with the similar property that the product of any two is one less than a rational square is known as a rational Diophantine m-tuple.
The question of existence of (integer) Diophantine quintuples was one of the oldest outstanding unsolved problems in Number Theory. In 2004 Andrej Dujella showed that at most a finite number of Diophantine quintuples exist. In 2016 a resolution was proposed by He, Togbé and Ziegler, subject to peer-review.
We can have a special case, where a Diophantine m-tuple is a set of m positive integers such that the product of any two of them increased by one unit is a perfect square, in other words the following set of numbers {1, 3, 8,} is a Diophantine triple because it satisfies the following conditions:
1 ∙3+1= 2^2
1 ∙8+1=3^2
3 ∙8+1=5^2
The study of Diophantine m-tuples dates back to the year 300 BC when the Greek mathematician Diophantus discovered the following set of rational numbers having the property described earlier: {1/16 ,33/16 ,17/4 ,105/16}.
This kind of special case where the product of the numbers is increased by one has been studied by a lot of people in the field of Diophantine analysis.
Other cases have been studied by mathematicians. For example, in the case of n ≠ 1, the following set {1, 2, 5} is Diophantine triple for the case D(-1). This set has proven to not be able to be extended any further. It was proven first by Brown in 1985 [3], then by Walsh [4] and Kihel [5] in 1999 and 2000. Another D(-1) triple proven to not be able to be extended to a D(-1) is {1, 5, 10}. This was proven by Mohanty and Ramasamy in 1984 [6].
In more recent years it has been proven that the sets {1, 2, 5} and {1, 5, 10} also cannot be extended to quadruples using Lucas and Fibonacci numbers.
Andrej Dujella, another important name in the field of Diophantine equations, proved in 1998 that all sets that take the form {1, 2, c} cannot be extended [8]. Later on it was also proven the nonextendability of the set {1, 5, c} by Muriefah and Al-Rashed [9], and in 2005, Filipin proved the set {1, 10, c} cannot be extended [10].
Definition and conditions for Diophantine triples
Definition 1.1. A set of three positive integers a, b, c where a < b < c, {a, b, c} is called a Diophantine triple, having the property D (n), where n is an integer different than 0, if there exist three integers q_1^2,q_2^2,q_3^2 that satisfy the following conditions:
q_1^2=ab+n
q_2^2=ac+n
q_3^2=bc+n
(6.1)
Example 1.1.
Consider the following: {a, b, c} a Diophantine triple equal to {1, 24, 35}, having n = 1 and (q_1^2 ,q_2^2 ,q_3^2) = (5, 6, 39).
This equation is true because of the following:
q_1^2=5^2=1×24+1=ab+n
q_2^2=6^2=1×35+1=ac+n
q_3^2=〖35〗^2=24×35+1=bc+n
The program used to find this triple can be found in the Appendix.
We can also have n be a negative number.
Consider the following: {a, b, c} a Diophantine triple equal to {1, 304, 309}, having n = -300 and (q_1^2 ,q_2^2 ,q_3^2) = (2, 3, 306).
The equation is true because of the following:
q_1^2=2^2=1×304-300=ab+n
q_2^2=3^2=1×309-300=ac+n
q_3^2=〖306〗^2=304×309-300=bc+n
The program used to find this triple can be found in the Appendix.
Remark 1.1. If we have a positive set of integers {q_1^2,q_2^2,q_3^2} have the property that q_1^2<q_2^2<q_3^2, there can exist more that one Diophantine triple that satisfies the equation (6.1). In layman’s terms, we can have more than one n satisfying (6.1).
Theorem 1.1. Consider a Diophantine triple {a, b, c}. If a+b+c is odd and a<b<c
Then {a, b, c} is a triple with D(n) property having:
q_1^2=1/2(a+b-c)
q_2^2=1/2 (c+a-b) and n=1/4(a^2+b^2+c^2-2ab-2ac-2bc)
q_3^2=1/2(b+c-a)
(9.1)
Proof. We know that a+b+c is even, then a+b-c, c+a-b and b+c-a are also even. Then, q_1^2 , q_2^2 , q_3^2 are all integers.
Let n=1/4(a^2+b^2+c^2-2ab-2ac-2bc), then ab+n=ab+1/4 (a^2+b^2+c^2-2ab-2ac-2bc)=〖[ 1/2(a+b-c)]〗^2=〖q_1^2〗^2
We can follow the same steps and get ac+n=〖q_2^2〗^2 and bc+n=〖q_3^2〗^2.
Now we can say about {a, b, c} that it is a Diophantine triples with D(n) property.
Diophantine quadruples
The first Diophantine quadruple was found by Fermat, and it is: {1, 3, 8, 120.}. Diophantus also found a set of rational quadruples. It was later proved by Baker and Davenport that a fifth positive integer cannot be added to this set. Euler was able to extend this set by adding the following rational number 777480/8288641. He also proved that any two elements in the set increased by one is a perfect square of a rational number. What is more, he found the following infinite family of quadruples {a,b,a+b+2r,4r(r+a)(r+b)}, if ab+1=r^2. In 1993, Andrej Dujella proved that if an integer n isn’t of form n=4k+2 and if n isn’t from the set {-4, -3, -1, 3, 5, 8, 12, 20}, then there is at least one Diophantine quadruple with the property D(n)[11].
The set {1, 3,8, 120} is a Diophantine quadruple because it satisfies the following conditions:
1 ∙3+1= 2^2
1 ∙8+1= 3^2
1 ∙120+1= 〖11〗^2
3 ∙8+1= 5^2
3 ∙120+1= 〖19〗^2
8 ∙120+1= 〖31〗^2
Definition and conditions for Diophantine quadruples
Definition 2.1. A set of four positive integers a, b, c, d where a < b < c < d, {a, b, c, d} is called a Diophantine quadruple, having the property D (n), where n is a integer different than 0, if there exist six positive integers q_1^2,q_2^2,q_3^2,q_4^2,q_5^2,q_6^2 that satisfy the following conditions:
q_1^2=ab+n
q_2^2=ac+n
q_3^2=ad+n
q_4^2=bc+n
q_5^2=bd+n
q_6^2=cd+n
(7.1)
Example 2.1. Consider the following: {a, b, c, d} a Diophantine quadruple equal to {2, 4, 12, 420} having n = 1 and (q_1^2 ,q_2^2 ,q_3^2 ,q_4^2 ,q_5^2 ,q_6^2) = (3, 5, 29, 7, 41, 71).
The equation is true because of the following:
q_1^2=3^2=2×4+1=ab+n
q_2^2=5^2=2×12+1=ac+n
q_3^2=〖29〗^2=2×420+1=ad+n
q_4^2=7^2=4×12+1=bc+n
q_5^2=〖41〗^2=4×420+1=bd+n
q_6^2=〖71〗^2=12×420+1=cd+n
The program used to find this quadruple can be found in the Appendix.
Diophantine quintuples
One of the oldest unsolved problems in number theory was whether there exists any Diophantine quintuple. In 2004 Andrej Dujella proved that at most there is a finite number of quintuples. In 2016 He, Togbe and Ziegler have claimed to prove that there are no Diophantine quintuples. We know from Mihai Cipu and Tim Trudgian that there are at most 1.18×〖10〗^27 Diophantine quintuples. Furthermore, every triple of form {a, b, c} can be extended to what is called a regular quadruple. If a Diophantine double or triples cannot be extended to a non-regular quadruple, then it cannot be extended to a quintuple.
Definition and conditions for Diophantine quintuples:
Definition 3.1. A set of five positive integers a, b, c, d, e where a < b < c < d < e, {a, b, c, d, e} is called a Diophantine quintuple, having the property D (n), where n is a integer different than 0, if there exist 10 integers q_1^2,q_2^2,q_3^2,q_4^2,q_5^2,q_6^2,q_7^2,q_8^2,q_9^2.q_10^2 satisfying the following conditions:
q_1^2=ab+n
q_2^2=ac+n
q_3^2=ad+n
q_4^2=ae+n
q_5^2=bc+n
q_6^2=bd+n
q_7^2=be+n
q_8^2=cd+n
q_9^2=ce+n
q_10^2=de+n
(8.1)
There is no known Diophantine quintuple having the property D(1). However we do know the smallest n = 256 for which there exists the following quintuples: {1, 33, 105, 320, 18240} and {5, 21, 64, 285, 6720}.
Example 3.1. Consider the following {a, b, c, d, e} a Diophantine quintuple equal to {5, 21, 64, 285, 6720} having n = 256 and (q_1^2 ,q_2^2 ,q_3^2 ,q_4^2 ,q_5^2 ,q_6^2,q_7^2 ,q_8^2 ,q_9^2 ,q_10^2 ) =(19, 24, 41, 184, 40, 79, 376, 136, 656, 1384).
The equation is true because of the following:
q_1^2=〖19〗^2=5×21+256=ab+n
q_2^2=〖24〗^2=5×64+256=ac+n
q_3^2=〖41〗^2=5×285+256=ad+n
q_4^2=〖184〗^2=5×6720+256=ae+n
q_5^2=〖40〗^2=21×64+256=bc+n
q_6^2=〖79〗^2=21×285+256=bd+n
q_7^2=〖376〗^2=21×6720+256=be+n
q_8^2=〖136〗^2=64×285+256=cd+n
q_3^2=〖656〗^2=64×6720+256=ce+n
q_4^2=〖1384〗^2=285×6720+256=de+n
Almost Diophantine quintuples
We understand an almost Diophantine quintuple with D(n) property, the set {a, b, c, d, e} a Diophantine quintuple, where a, b, c, d, e are integers and a < b < c < d < e, for which the set {a, b, c, d} is a Diophantine quadruples with D(n) property and out of the following conditions: ae+n, be+n, ce+n, de+n being perfect squares, only 3 are true.
We can talk about the best or most accurate almost Diophantine quintuples having the D(n) property on a random search domain.
Let x be from the set {a, b, c, d, e} for which xe+n is not a perfect square. Then we can say about the set {a, b, c, d, e} that it is the best quintuple having almost D(n) property, if the relative error between xe+n and the closest perfect square is the smallest on the respective search domain. In other words, xe+n is not a perfect square but is the closest to a perfect square on that search domain. Cira and Dujella have found 31 Diophantine quintuples with almost D(1) property on the following search domain: {1≤a≤40,a<b≤500,b<c<25×〖10〗^3,c<d≤15×〖10〗^4,d<e≤〖10〗^6}.
Chapter 2
Solving Diophantine equations
Analytical solving
By analytical solving we imply finding a general solution that completely solves the Diophantine equation.
In 1990 David Hilbert proposed a list of mathematical problems. The 10th one asked if there exists an algorithm that can determine whether a Diophantine equation has solutions. Such an algorithm exists only for solutions of first order Diophantine equations. It was later proven that it is impossible to find a general solution.
Euclidean algorithm
One of the prerequisites of solving Diophantine equations is understanding the Euclidean algorithm. Here I will explain how it works and give an example.
The Euclidean algorithm, named after the ancient Greek mathematician Euclid, who first published it in 300 BC, is one of the oldest and well-known algorithms in mathematics. We use the Euclidean algorithm to compute the greatest common divisor (gcd) of two integers a and b. It also serves as a foundation for more complex algorithms used in number theory.
The Euclidean algorithm is basically a continual repetition of the division algorithm for integers. We basically repeatedly divide the divisor by the until the remainder is 0.
Definition 4.1 If a and b are positive integers, there exist unique non-negative integers q and r so that
a=qb+r,where 0≤r<b
q is called the quotient and r the remainder.
The greatest common divisor is the last non-zero remainder in the algorithm. Let’s look at the example below to find the greatest common divisor between 102 and 38.
Example 4.1 Find the greatest common divisor between 102 and 38
102=2×38+26
38=1×26+12
26=2×12+2
12=6×2+0
The greatest common divisor is 2 because it is the last non-zero remainder that appears before the algorithm terminates.
Recursively implementing the Euclidean algorithm is fairly straight forward.
int greatestCommonDivisor(int m, int n)
{
if(n == 0) return m;
return greatestCommonDivisor(n, m % n);
}
Linear Diophantine equations
A linear Diophantine equation is of form ax+by=c and may have many solutions or no solutions at all.
In order to find solutions to a linear Diophantine equation firstly we need to identify an initial solution that we then alter to obtain the remaining solutions.
First, we need to know if there even exist any solutions. We can do this by using Bezout’s Identity, which states the following:
Let a and b be two integers different from 0 and d=cgd(a,b). Then there exist integers x and y that satisfy
ax+by=d
Furthermore, there exist integers x and y satisfying
ax+by=c
if and only if c divides n.
So, if a, b, c are integers, where a and b are both different than 0, then the linear Diophantine equation ax+by=c has an integral solution if and only if gcd(a, b) is a divisor of c.
Example 5.1 Find integral solutions to the linear Diophantine equation
14x+91y=53
First we compute the gcd(14, 91) = 7. Then, we see that 7 does not divide 53, therefore by Bezout’s Identity we know that there are no integer solution to the equation above.
On the other hand, if there exist solutions to our equation an well-structured algorithm has been devised in order to solve the equation.
The technique used in order to find initial solutions to the linear Diophantine equation is explained bellow.
If we have the following Diophantine equation:
ax+by=c.
First we will find the greatest common divisor of a and b using the Euclidean algorithm and set it equal to d.
If d does not divide c there are no solution, otherwise continue to the next step.
Reformat the equation from the Euclidean algorithm.
Using substitution, go through the steps of the Euclidean algorithm to find a solution to the equation ax_i+by_i=d.
The initial solution to our equation ax+by=c is the ordered pair (x_i⋅c/d,y_i⋅c/d).
Example 6.1. Find initial integer solutions to the equation
141x+34y=30
First using the Euclidean algorithm, we obtain
141=4(34)+5
34=6(5)+4
5=1(4)+1
So, the greatest common divisor between 141 and 34 is 1. Because 1 divides 30 we know that solutions exist to the equation.
Now, we reformat the equation from the Euclidean algorithm, obtaining
5=141-4(34)
4=34-6(5)
1=5=-1(4)
After this we use substitution in order to find a solution to the equation
141x_i+34y_1=1
1=5-1(4)
1=5-1[34-6(5)]
1=7(5)-1(34)
1=7[141-4(34)]-1(34)
1=7(141)-29(34)
From this is results that x_i=7 and y_i=-29 are solutions to the equation 141x_i+34y_1=1.
Therefore, the initial solution to the equation 141x+34y=30 is
x=7×30=210
y=-29×30=-870
Now I will show how we can find a general solution to this equation.
We have found only one solution to the equation above. When the equation ax+by=c has integer solutions, there exist infinite solutions.
Theorem 1.1. If we have a linear Diophantine equation ax+by=c, having integer solutions of the form (x^*,y^*), then all integer solutions to the equation are of the form
(x^*+m b/gcd(a,b) ,y^*-m a/gcd(a,b) )
where m is an integer.
Proof.
We have
a(x^*+m b/gcd(a,b) )+b(y^*-m a/gcd(a,b) )=ax^*+by^*+abm/gcd(a,b) -abm/gcd(a,b)
=ax^*+by^*
=c
This shows that these are the solutions to the Diophantine equation. Now, given any solution (x,y) we have
ax+by=〖ax〗^*+by^*
a(x-x^* )=-b(y-y^*)
a/gcd(a,b) (x-x^* )=-b/gcd(a,b) (y-y^* ).
Because a/gcd(a,b) and b/gcd(a,b) are relatively prime, there exist an integer m such that x-x^*=m b/gcd(a,b) and y-y^*=-m a/gcd(a,b) , which proves the theorem above.
Empirical solution
By an empirical solution we need to find an algorithm that determines the solution for the Diophantine equation on a finite domain.
The first theorem proven using a computer was the four color theorem, and it was proved by Kenneth Appel and Wolfgang Haken in 1976.
In order to find solutions to the Diophantine equations, we will use a program written in Mathcad.
First, let’s start by explaining what each of the functions in Mathcad do.
ORIGIN: The ORIGIN variable is the variable that sets the indexing for matrixes. When the user sets the ORIGIN to a value of 1, Mathcad should start the indexing of matrixes at one.
round (z, n): Rounds z to n places. If n is omitted, z is rounded to the nearest integer. If n < 0, z is rounded to the left of the decimal point.
stack (A, B, C, …): Returns an array formed by placing A, B, C, … top to bottom. A, B, C, … are arrays having the same number of columns, or they are scalars and column vectors.
Partial solving of Diophantine triples
Consider a Diophantine triple {a, b, c} having D(n) property.
Program 1.1
We will use the program to find Diophantine triples having D(1) property.
The program for solving this equation can be found in the Appendix.
The call of the program is done by the sequence:
af=10,bf=〖10〗^3,cf=〖10〗^5,o=2,n=1
Therefore, we have the following search domain:
1≤a≤10,a<b≤〖10〗^3,b<c≤〖10〗^5
The call of the program:
t_0:time(0),S≔P3(af,bf,cf,o,n)
The execution time in seconds and the number of solutions:
(t_1-t_o )∙sec=”0:0:8.917″∙hhmmss nrrows≔rows(S)-1
nrrows=399
So, the program executed for 8.917 seconds and we have found 399 solutions.
The program found in Appendix displays only the first 40 triples and solutions.
The first Diophantine triple found was {1,3,8} and the last one was {10,980,1.188×〖10〗^3}.
Program 2.1.
We will use the program to find Diophantine triples having D(-300) property.
The program for solving this equation can be found in the Appendix.
The call of the program is done by the sequence:
af=10,bf=〖10〗^3,cf=〖10〗^5,o=2,n=-300
Therefore, we have the following search domain:
1≤a≤10,a<b≤〖10〗^3,b<c≤〖10〗^5
The call of the program:
t_0:time(0),S≔P3(af,bf,cf,o,n)
The execution time in seconds and the number of solutions:
(t_1-t_o )∙sec=”0:0:8.917″∙hhmmss nrrows≔rows(S)-1
nrrows=682
So, the program executed for 8.917 seconds and we have found 399 solutions.
The program found in Appendix displays only the first 39 triples and solutions.
The first Diophantine triple found was {1,75,304} and the last one was {10,840,1.03×〖10〗^3}.
Partial solving of Diophantine quadruples
Consider a Diophantine quadruple {a, b, c, d} having D(n) property.
Program 3.1
We will use the program to find Diophantine quadruples having D(1) property.
The program for solving this equation can be found in the Appendix.
The call of the program is done by the sequence:
af=10,bf=〖10〗^2,cf=〖10〗^3,〖df=20〗^4,o=2,n=1
Therefore, we have the following search domain:
1≤a≤10,a<b≤〖10〗^2,b<c≤〖10〗^3,c<d≤〖20〗^4
The call of the program:
t_0:time(0),S≔P1(af,bf,cf,df,o,n)
The execution time in seconds and the number of solutions:
(t_1-t_o )∙sec=”0:18:5.069″∙hhmmss rownr≔rows(S)-1
townr=64
So, the program executed for 18 minutes, 5 seconds and we have found 64 solutions.
The first Diophantine quadruple found was {1,3,8,120} and the last one was {10,36,84,121220}.
Program 4.1
We will use the program to find Diophantine quadruples having D(-25) property.
The program for solving this equation can be found in the Appendix.
The call of the program is done by the sequence:
af=10,bf=〖10〗^2,cf=〖10〗^3,〖df=20〗^4,o=2,n=-300
Therefore, we have the following search domain:
1≤a≤10,a<b≤〖10〗^2,b<c≤〖10〗^3,c<d≤〖20〗^4
The call of the program:
t_0:time(0),S≔P1(af,bf,cf,df,o,n)
The execution time in seconds and the number of solutions:
(t_1-t_o )∙sec=”0:18:5.069″∙hhmmss nrrows≔rows(S)-1
nrrows=11
So, the program executed for 18 minutes and 5 seconds and we have found 11 solutions.
The first Diophantine quadruple found was {3,52,100,292} and the last one was {7,100,228,628}.
Partial solving of Diophantine quintuples
Conclusions
The work that Diophantus carried out in the 3rd century AD has reverberated across centuries and is even talked about in the modern era of mathematics. Diophantine equations have been studied for more than 3000 years yet there still remain problems that have not been solved. This is what led me to elaborate this Bachelor’s Thesis where I have tried to lay a basis for the understanding of Diophantine equations.
We have seen throughout the course of this Bachelor’s Thesis a little history of the Diophantine equation and the mathematicians that influenced this field, we have seen examples of equations and the methods used for solving them both analytically and empirically.
Diophantine equations are not an abstract concept, that concern only mathematicians hence why they should not be dismissed easily. They have real world applications especially in coding theory and cryptography. For example, elliptic curve cryptography is based on doing calculations in finite field (also called Galois fields) for a Diophantine equation of degree 3 in two variables.
The programs that have been presented here can be perhaps used in the future, when greater computing power will be available, and many more solutions can be found, in a much shorter period of time.
As the computing power is increasing every year it is more and more obvious that computers are becoming an essential tool in solving problem in the field of number theory. Despite this fact, there are still problems that require very long computing, for example finding Diophantine quadruples on a large search domain can take hours, days and even weeks.
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