# On Squarimes (Square Primes)

In this note I have mentioned that it is not possible that $$p_1^2+p_2^2=p_3^2$$ where $$p_1,p_2,p_3$$ are all primes

I have spend some more time on it and got another proof for it which I consider the best one:

For $p_1,p_2,p_3$ all of which are primes it is not possible to have $p_1^2+p_2^2=p_3^2$

Proof:

Let us assume there exists primes $p_1,p_2,p_3 | p_1^2+p_2^2=p_3^2$ $p_1^2=p_3^2-p_2^2=(p_3-p_2)(p_3+p_2)$ Now pairwise factors of $p_1^2$ are $(p_1^2,1),(p_1,p_1)$ now if $p_3-p_2=p_1\Rightarrow p_3-p_2=p_1=p_3+p_2\Rightarrow p_2=0$ which is not possible $\therefore p_3-p_2=1;p_3+p_2=p_1^2 \because p_3-p_2 But this also leads to contradiction because according to it $p_3=p_2+1$ which is not possible for $p_3>3$

Therefore no solution exists for $p_3>3$ if $p_3=3\Rightarrow p_2=2$ but $p_3^2-p_2^2=3^2-2^2=5 \Rightarrow p_1^2=5 \Rightarrow p_1=\sqrt{5},-\sqrt{5}$ which also leads to contradiction

Therefore no squarime exists

Note by Zakir Husain
10 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.

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: