# Only Finite Integer Solutions to These Quartics

Some guidance questions before you tackle Diophantine Quartic. If you can manage to prove these, you will be able to solve my question with no problem.

Problem 1: Prove that for all positive ordered pairs of integers $(x,y)$, there does not exist an integer solution to $x^4+x+1=y^2$

Problem 2: Prove that for all positive ordered pairs of integers $(x,y)$, there does not exist an integer solution to $x^4+x+2=y^2$ other than the pair $(1,2)$

Problem 3: Prove that for all positive ordered pairs of integers $(x,y)$, there does not exist an integer solution to $x^4+x+7=y^2$ other than the pairs $(1,3)$ and $(2,5)$

Post your solutions below. Have fun!

Note by Daniel Liu
7 years, 2 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
• Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

Problem 1: Note that $x$ is given to be positive. Hence, the given equation implies that $y^2 > x^4$, i.e. $y> x^2$, i.e. $y \geq (x^2+1)$. Hence $y^2\geq x^4 + 2x^2+1$. The given condition then implies $x^4 + x +1 \geq x^4 + 2x^2 +1$, i.e. $x \geq 2x^2$. which does not have any positive integral solution. Other two problems can also be nailed down by similar arguments.

- 7 years, 2 months ago

Awesome!!I also used similar arguments...

- 7 years, 2 months ago

Yep, that's how I solved it.

- 7 years, 2 months ago

[Whispers with ghostly voice] ...bound between squares...

- 7 years, 2 months ago

Correct!

- 7 years, 2 months ago

I would like to share a few more problems which use similar ideas.

- 7 years, 2 months ago

We are given that y^2 > x^4+x+1.............. y^2 > x^4.............. y > x^2................. It is also given that x is a positive integer............. So, y>=x^2+1.............
y^2>=x^4+2x^2+1......................
So, y can not be equal to x^4+x+1.....................

- 7 years, 2 months ago