Question

Either find all solutions or prove that there are no solutions for the diophantine equation \(2 x+14 y=17\).

Short answer

If so, find them. Answer: No, there are no solutions for the diophantine equation \(2x + 14y = 17\).

Step-by-step solution

Find the gcd of coefficients and check if it divides 17

First, we need to find the gcd of 2 and 14, that is, gcd(2, 14) = 2. Since 2 does not divide 17, we can conclude that there are no solutions for the given diophantine equation.

Conclude that there are no solutions

Based on Step 1, we can now conclude that there are no solutions for the diophantine equation \(2x + 14y = 17\).

Key concepts

Greatest Common Divisor

The Greatest Common Divisor, often abbreviated as GCD, is a fundamental concept in number theory. Put simply, the GCD of two numbers is the largest positive integer that divides both numbers without leaving a remainder.
To find the GCD, one common method is to use prime factorization where each number is expressed as a product of prime numbers, then the GCD is the product of the lowest powers of all common primes. Alternatively, the Euclidean algorithm is more efficient, especially for larger numbers. This method involves dividing the larger number by the smaller one, taking the remainder, and repeating the process using the original smaller number and this remainder until the remainder is zero. The last non-zero remainder is the GCD. For instance, the GCD of 2 and 14 can quickly be found by recognizing that they are both multiples of 2.
Understanding the GCD is crucial when solving Diophantine equations because it allows us to assess the possibility of integer solutions. If the GCD of the coefficients does not divide the constant term in the equation, as was the case with the equation \(2x + 14y = 17\), it immediately indicates that there are no integer solutions.

Integer Solutions

In mathematics, specifically number theory, integer solutions refer to solutions of equations that are whole numbers. When dealing with Diophantine equations like \(2x + 14y = 17\), we are particularly concerned with finding such solutions.
Diophantine equations are often linear and can be written in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are integers. To determine if integer solutions exist, we check the divisibility of \(c\) by the GCD of \(a\) and \(b\). If the constant on the right side of the equation (here, 17) is divisible by the GCD of the coefficients of \(x\) and \(y\) (here, 2), then integer solutions exist. If not, as with our problem, finding integer solutions is impossible.
This property is a consequence of a more general principle in number theory: if a linear Diophantine equation has solutions, then an infinite number of them exist, differing by multiples of certain constants derived from the equation's coefficients.

Number Theory

Number theory is a branch of mathematics devoted to the study of integers and integer-valued functions. It encompasses a wide range of topics, including prime numbers, divisibility, and the properties of numbers.
One of the critical applications of number theory is solving Diophantine equations, which are equations that demand integer solutions. The study of such equations dates back to ancient Greece and India with significant contributions from figures like Diophantus and later, Pierre de Fermat.
In our specific example involving the equation \(2x + 14y = 17\), number theory offers the tools, such as the concept of GCD and divisibility rules, to determine if solutions exist even before attempting to find explicit values for \(x\) and \(y\). Understanding these foundational concepts enriches our ability to explore more complex mathematical ideas and solve seemingly challenging problems by breaking them down into simpler arithmetic properties.