Sophie Germain’s identity(divisibility,number theory)

The most useful formula in competitions is the fact that abanbna-b | a^n-b^n for all n, and a+ban+bna+b | a^n+b^n for odd n.We have a2b2=(ab)(a+b)a^2-b^2=(a-b)(a+b). But a sum of two squares such as x2+y2x^2 + y^2 can only be factored if 2xy is also a square. Here you must add and subtract 2xy. The simplest example is the identity of Sophie Germain:

a4+4b4=a4+4a2.b2+4b44a2.b2=(a2+2b2)2(2ab)2=(a2+2b2+2ab)(a2+2b22ab)a^4 + 4b^4 = a^4 + 4a^2.b^2 + 4b4 - 4a^2.b^2 = (a^2 + 2b^2)2 - (2ab)^2 = (a^2 + 2b^2 + 2ab)(a^2 + 2b^2 - 2ab)

Some difficult Olympiad problems are based on this identity. For instance, in the 1978 Kurschak Competition, we find the following problem which few students solved.

example:1 n>1n4+4nn > 1 ⇒ n^4 + 4^n is never a prime. If n is even, then n4+4nn^4 +4^n is even and larger than 2. Thus it is not a prime. So we need to show the assertion only for odd n. But for odd n=2k+1n = 2k + 1, we can make the following transformation, getting Sophie Germain’s identity: n4+4n=n4+442k=n4+4(2k)4n^4 + 4^n = n^4 + 4·4^{2k} = n^4 + 4 · (2k)^4 which has the form a4+4b4a^4 + 4b^4. This problem first appeared in the Mathematics Magazine 1950. It was proposed by A. Makowski, a leader of the Polish IMO-team. Quite recently, the following problem was posed in a Russian Olympiad for 8th graders:

Note by Chakravarthy B
7 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

Ok. I changed it.

chakravarthy b - 7 months, 2 weeks ago

Log in to reply

No you didn't. It is still the same.......

Aaghaz Mahajan - 7 months, 2 weeks ago

Log in to reply

Once check

chakravarthy b - 7 months, 2 weeks ago

Log in to reply

@Chakravarthy B Yup, now it is fine....!!

Aaghaz Mahajan - 7 months, 2 weeks ago

Log in to reply

Did anyone understand this?

chakravarthy b - 7 months, 2 weeks ago

Log in to reply

Yeah I did........ although, there is a typo.....The question should be n4+4nn^4+4^n instead of n4+4nn^4+4n

Aaghaz Mahajan - 7 months, 2 weeks ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...