Inspired by Kalpok Guha

I happened to see this problem. So I came up with a similar problem below.


How many integers n>1n>1 are there such that n,n+2,n+6n,n+2,n+6 are all prime numbers ?

Do post Awesome solutions with proofs!

Note by Nihar Mahajan
4 years, 4 months ago

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My intuition says there are infinitely many such nn.

Nihar Mahajan - 4 years, 4 months ago

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If there are infinitely many such n, then the twin prime conjecture is true. I suggest working on the twin prime conjecture first, because it is (likely) easier.

Calvin Lin Staff - 4 years, 4 months ago

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Yeah , I also think that.Thanks!

Nihar Mahajan - 4 years, 4 months ago

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Has anyone come close to proving the twin prime conjecture or do we need to solve the Riemann hypothesis first?

Sharky Kesa - 4 years, 4 months ago

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@Sharky Kesa Zhang Yitang has proved a weakened form of the Twin Prime conjecture, namely that there exists an N such that there are infinitely many pairs of consecutive primes with difference < N.

Here is a good introductory read to the relationships between TP and RH. It is written by Dan Goldston (same as the above), though prior to Zhang's discovery. If you can understand through the first 5 chapters, that would be great

The conjecture that the distribution of twin primes satisfies a Riemann Hypothesis type error term is well supported empirically, but I think this might be a problem that survives the current millennium.

Calvin Lin Staff - 4 years, 4 months ago

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Yes, even as I was working out I found many numbers,

Here's my explanation,

  • First take prime numbers ending with 11,

you can find that there can be many triplets satisfying the condition. For example: 11,13,17\boxed{ 11, 13, 17}, 41,43,47\boxed{41,43,47}, 101,103,107\boxed{101,103,107} etc. . .

  • Now, let's take primes ending with 22,

As there is only one possibility, but that proves to be wrong. 2,4,8\boxed{2,4,8}

  • Now, let's take primes ending with 33,

We can say there's only no such possibility, because the second number i.e. n+2n+2 yields us a number divisible by 55. Even this triplet proves wrong, 3,5,9\boxed{3,5,9} as 99 is not prime.

  • Now let's take primes ending with 55,

Only one possibility, i.e. 5,7,11\boxed{5,7,11}. In all other triplets, the first number i.e. nn is not prime

  • Now, let's take primes ending with 77,

You can find that the can be many possibilities. For example: 17,19,23\boxed{ 17, 19, 23}, 107,109,113\boxed{107,109,113}, 227,229,233\boxed{227,229,233} etc. . .

  • Now, let's take primes ending with 99,

No such possibility, because the third number i.e. n+6n+6 yields us a number divisivisible by 55

Sravanth Chebrolu - 4 years, 4 months ago

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I didn't know 71+6=79. :P

Sharky Kesa - 4 years, 4 months ago

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@Sravanth Chebrolu I think you must develop a habit of always cross checking your work. :)

Nihar Mahajan - 4 years, 4 months ago

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Lol Btw Nice observation.

Aditya Chauhan - 4 years, 4 months ago

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197+6=203

Aditya Chauhan - 4 years, 4 months ago

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Where did you comment the 3 comments? I can only see one here.

pic pic

Nihar Mahajan - 4 years, 4 months ago

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Ah! Those were deleted by me :P

Sravanth Chebrolu - 4 years, 4 months ago

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@Sravanth Chebrolu I guess U r the fastest one to get 200k points. Keep solving and making more problems.

Aditya Chauhan - 4 years, 4 months ago

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@Aditya Chauhan Thanks! I'll be posting a new set keep your eyes on that!

Sravanth Chebrolu - 4 years, 4 months ago

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