Sophie Germain’s identity(divisibility,number theory)

The most useful formula in competitions is the fact that \[a−b | a^n−b^n\] for all n, and \[a+b | a^n+b^n\] for odd n.We have \[a^2−b^2=(a−b)(a+b)\]. But a sum of two squares such as \[x^2 + y^2\] 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:

\[a^4 + 4b^4 = a^4 + 4a^2.b^2 + 4b4 − 4a^2.b^2 = (a^2 + 2b^2)2 − (2ab)^2 = (a^2 + 2b^2 + 2ab)(a^2 + 2b^2 − 2ab)\]

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 \[n > 1 ⇒ n^4 + 4^n\] is never a prime. If n is even, then \[n^4 +4^n\] 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 \[n = 2k + 1\], we can make the following transformation, getting Sophie Germain’s identity: \[n^4 + 4^n = n^4 + 4·4^{2k} = n^4 + 4 · (2k)^4\] which has the form \[a^4 + 4b^4\]. 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
1 month, 3 weeks ago

No vote yet
1 vote

  Easy Math Editor

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 \( 2 \times 3 \)
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} \( \boxed{123} \)

Comments

Sort by:

Top Newest

Did anyone understand this?

chakravarthy b - 1 month, 3 weeks ago

Log in to reply

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

Aaghaz Mahajan - 1 month, 2 weeks ago

Log in to reply

Ok. I changed it.

chakravarthy b - 1 month, 2 weeks ago

Log in to reply

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

Aaghaz Mahajan - 1 month, 2 weeks ago

Log in to reply

Once check

chakravarthy b - 1 month, 2 weeks ago

Log in to reply

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

Aaghaz Mahajan - 1 month, 2 weeks ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...