HELP ME TO SOLVE THIS!!

Find all integral solution of x4+y4+z4-w4=1995. Here, x4 means 'x raise to the power 4.

Note by Shu Ary
8 months, 2 weeks ago

No vote yet
1 vote

  Easy Math Editor

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. numbered
2. 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 1

paragraph 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 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

There are no integral solutions.............Hint :- Look mod 16.............

Aaghaz Mahajan - 8 months, 2 weeks ago

Log in to reply

How!! Didn't understand?Please elaborate it!!

Shu Ary - 8 months, 2 weeks ago

Log in to reply

x4 is congruent to either 0 or 1 mod 5.............Now, the RHS is congruent to 0 mod 5, so x4+y4+z4 = w4 (mod5)............And so, this is only possible when x,y,z,w are all multiples of 5............but that is not possible or else 1995 would be divisible by 625.............Hence, there are no integral solutions

Aaghaz Mahajan - 8 months, 2 weeks ago

Log in to reply

@Aaghaz Mahajan For x4+y4+z4w4(mod5) x^4+y^4+z^4 \equiv w^4 \pmod 5 , there's also the possibility 1+0+01(mod5) 1+0+0 \equiv 1 \pmod 5 .

Henry U - 8 months, 2 weeks ago

Log in to reply

@Henry U Yeah you are right.............But I figured it out.......mod 5 is not a good option to look...........instead, try mod 16............that works fine.......:)

Aaghaz Mahajan - 8 months, 2 weeks ago

Log in to reply

@Aaghaz Mahajan Didn't understand your last line??

Shu Ary - 8 months, 2 weeks ago

Log in to reply

@Shu Ary I'm sorry I screwed up.........mod 5 is not a good option.......look mod 16 instead........that works........

Aaghaz Mahajan - 8 months, 2 weeks ago

Log in to reply

@Aaghaz Mahajan Thank you!!!

Shu Ary - 8 months, 2 weeks ago

Log in to reply

@Shu Ary You are welcome........!!

Aaghaz Mahajan - 8 months, 2 weeks ago

Log in to reply

Any number to the power of 4 has a remainder of 0 or 1 when divided by 16.

So x4+y4+z4w4x^4 + y^4 +z^4 - w^4 can only have a remainder of 0,1,2, or 3 when divided by sixteen.

When 1995 is divided by 16 it leaves a remainder of 11.

Therefore there are no integral solutions.

Chris Sapiano - 4 months, 2 weeks ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...