Show that the equation $2a^3 - 7b^2 = 1$ has no solution over the integers

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In summary, an equation has no solution over the integers when there are no values for the variables that satisfy the equation when restricted to only integer values. This can be proven using proof by contradiction. The equation $2a^3 - 7b^2 = 1$ is an example of a Diophantine equation, often used in number theory and applications in cryptography and coding theory. While an equation can have no integer solutions, it can still have solutions over other sets of numbers, such as rational or real numbers. And even when the variables are not restricted to integers, the equation $2a^3 - 7b^2 = 1$ can still have solutions, such as fractions or decimals. However, it remains
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Euge
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Here is this week's POTW:

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Show that the equation $2a^3 - 7b^2 = 1$ has no solution over the integers.

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No one answered this week's problem. You can read my solution below.
Reducing the equation modulo $7$ yields $2a^3 \equiv 1 \pmod{7}$, or $a^3 \equiv 4 \pmod{7}$. Note that $\pm 1, \pm 2$, and $\pm 3$ all have cubes that are $\equiv \pm 1\pmod{7}$. Hence, the congruence $a^3 \equiv 3\pmod{7}$ has no solution. This implies the original Diophantine equation has no solution.
 

Related to Show that the equation $2a^3 - 7b^2 = 1$ has no solution over the integers

1. What does it mean for an equation to have no solution over the integers?

Having no solution over the integers means that there are no possible values of the variables that will make the equation true when substituted into the equation. In other words, there are no whole numbers that can satisfy the equation.

2. How can you prove that the equation $2a^3 - 7b^2 = 1$ has no solution over the integers?

One way to prove this is by using the method of contradiction. Assume that there is a solution to the equation over the integers, then show that this leads to a contradiction or an impossibility. This would prove that the initial assumption was false, and therefore the equation has no solution over the integers.

3. Can the equation have a solution over the real numbers?

Yes, the equation may have solutions over the real numbers. In fact, there are infinitely many real solutions to this equation. However, the question specifically asks about solutions over the integers, which would exclude any non-whole number solutions.

4. Are there any other methods to prove that the equation has no solution over the integers?

Yes, there are other methods such as using modular arithmetic or number theory concepts. These methods may involve looking at the remainders of the numbers when divided by certain integers or using properties of prime numbers.

5. Why is it important to determine if an equation has no solution over the integers?

It is important to determine this because it helps us understand the behavior of the equation and its solutions. It also allows us to focus on finding solutions in other sets of numbers, such as the real numbers or complex numbers. Additionally, it can lead to the discovery of interesting mathematical patterns and relationships.

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