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
1 year, 4 months ago

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Did anyone understand this?

chakravarthy b - 1 year, 4 months ago

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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 - 1 year, 4 months ago

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Ok. I changed it.

chakravarthy b - 1 year, 4 months ago

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

Aaghaz Mahajan - 1 year, 4 months ago

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Once check

chakravarthy b - 1 year, 4 months ago

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

Aaghaz Mahajan - 1 year, 4 months ago

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Does this seem familiar. You can clearly see what you want to draw or paint but when you sit down to create an artwork you just can't capture it. It seems mysterious and wonderful how your favourite artists created such beauty. But I know all the truth about school art classes and that's why I recommend you to use this service https://essayreviewexpert.com/best-thesis-writing-service/ where you can find the best thesis writing services. They weren't superhuman prodigy's for the most part. They learned just as you can.

Sammie Oki - 5 months, 2 weeks ago

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