Here is a really neat proof of the infinitude of primes! It's so simple that it makes me say: "Why didn't I think of it?". In fact, it was made just in 2005, by Filip Saidak, and I was really surprised to know that. So here is the proof:

Let \(n > 1\) be a positive integer. Since \(n\) and \(n+1\) are consecutive integers, they must be coprime, and hence the number \(N_2 = n(n + 1)\) must have at least two different prime factors. Similarly, since the integers \(n(n+1)\) and \(n(n+1)+1\) are consecutive, and therefore coprime, the number \(N_3 = n(n + 1)[n(n + 1) + 1]\) must have at least 3 different prime factors. This can be continued indefinitely.

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## Comments

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TopNewestThat's really nice!!!

Darn I have to come up with something like that. :P

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A different proof also exists for the almost same result and with the same concept , apparently.

The proof goes as -

Why didn't I think of it !Log in to reply

Thas' a cool rearrangement o' what we learn in school....

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It seems that the proof shown is for infinitude of composite numbers. I fail to see how it shows that there are infinite number of primes,using this proof??

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if continued indefinitely, the proof is saying that there is a never ending list of composite numbers with a never ending list of unique prime factors. It is different than a proof by contradiction of a largest prime, and is more like a proof of an infinite number of prime factors

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the interesting thing of primes is that we could come with even simpler proofs of primes and extricating proofs too of primes simpler than that proof of infinite primes infinitude of primes by euclid and still after 2000 years gone by after euclid

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If \(\frac{n-1}{\phi(n)}\) is an integer then prove that no prime

`p`

gives integer value when \(\frac{n}{p^2}\).Log in to reply

That's really nice!!!

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I wish to know the proof of this geometric problem. Given a scalene triangle on the sides of which are drawn equilateral triangles having the sides of the given triangle as a side. Prove that the triangle formed by connecting the center of gravity of the three equilateral triangle is equilateral.

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I was unable to sow the proof of this geometric problem. I wish someone can help me.

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dang

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really

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:)

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Amazing!

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2^prime-1= prime..That means there r infinite prime no.s

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\(2^p-1\) is not always prime for \(p\) being prime. Eg : \(2^{11}-1=2047=23 \times 89\). So, your statement does not actually prove infinite number of primes.

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Euclid proves the same by induction over 2000 years ago

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Isn't it proof by contradiction ?

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