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Superprimes

I recently came across a type of numbers called as superprimes. A superprime is an integer (such as 7331) such that all its left-to-right initial segments are prime (for 7331 the segments are 7, 73, 733, and 7331, all prime).

The fun fact is, there is a largest possible superprime. Can you find it ?

Note by Bruce Wayne
3 years, 8 months ago

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I get 73939133. Patrick Corn · 3 years, 8 months ago

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@Patrick Corn Yup.The largest superprime is 73939133. Can you show how you arrived at this answer? Bruce Wayne · 3 years, 8 months ago

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@Patrick Corn how did u get it? Akshay Ginodia · 3 years, 8 months ago

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@Akshay Ginodia You need a computer program. Just generate the list \( L_n \) of superprimes with \( n \) digits (base case \( n =1 \), \( L_1 = \{ 3, 7 \} \)) and then use it to generate the list of superprimes with \( n+1 \) digits, by having your program check whether \( 10p + 1, 10p + 3, 10p + 7, 10p + 9 \) are prime, for all \( p \in L_n \). Repeat until \( L_n \) is empty. In particular I got that \( L_8 \) had two elements and \( L_9 \) had none. The prime 73939133 was the larger of the two. Patrick Corn · 3 years, 8 months ago

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@Patrick Corn Well, \(L_1=\{2,3,5,7\}\), so you are missing a few. I think that \(L_8\) has about five elements. Mark Hennings · 3 years, 8 months ago

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@Mark Hennings Yes indeed, thanks for pointing that out! I agree that \( |L_8| = 5 \). Patrick Corn · 3 years, 8 months ago

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@Patrick Corn elegant approach Gaurav Jain · 2 years, 5 months ago

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There is one more interesting thing called the 'EMIRP'.

Its a prime from both the ways(i.e from left to right and right to left).

Example; 13,17,31.37... Ranjana Kasangeri · 3 years, 8 months ago

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@Ranjana Kasangeri Is their any solid reason or proof for the 'statement' you stated. If their then please provide. Shubham Kumar · 3 years, 8 months ago

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@Shubham Kumar I was wrong,the problem is a conjecture

I thought it was obvious to have infinitely many emirps. Ranjana Kasangeri · 3 years, 8 months ago

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@Shubham Kumar If I had a proof of the infinitude of emirps I'd just say "There are infinitely many." If instead I said "There must be infinitely many" it would be a statement about how the world should be if there's any justice, rather than a statement of fact :).

Heuristically, one should expect infinitely many emirps: there are about \(O(10^d/d)\) primes with \(d\) digits, so about \(O(10^d/d^2)\) of them would be primes in reverse, as long as there are no unexpectedly negative correlations between primes and reverse primes. Erick Wong · 3 years, 8 months ago

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@Erick Wong The number of palindromic primes and the number of emirps (e.g. whether they're infinite or not) are both open problems. Michael Tong · 3 years, 8 months ago

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Your name is Bruce Wayne? Tanishq Aggarwal · 3 years, 8 months ago

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@Tanishq Aggarwal Nops :P. I just use this name every now and then. I have seen screen names like 'Harvey Dent' and 'Batman' too. Harvey Dent · 3 years, 8 months ago

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@Tanishq Aggarwal I don't think so! Ranjana Kasangeri · 3 years, 8 months ago

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Weeks ago I did this: http://archives.somee.com to search for prime numbers, I think I could easily add a method to search for superprimes. But I'd love to see someone solve this by pure mathematics. Mateo Torres · 3 years, 8 months ago

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I don't believe that there aren't an infinite amount of super primes... Yash Talekar · 3 years, 8 months ago

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@Yash Talekar Do you believe there is an infinite string of digits whose initial segments are always prime? (Hint: the two beliefs are equivalent.)

Heuristically, in any base \(b\) one should expect only finitely many superprimes: for any superprime of length \(n\) there are \(b\) possible extensions to length \(n+1\), and only about 1 in \(n \log b\) candidates of that size will be prime. Thus (heuristically) the superprimes will start to dwindle after passing \(n > b/\log b\) digits, until there are no more. Erick Wong · 3 years, 8 months ago

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How would anyone even hope to find the largest superprime without either using a computer program or just searching it up? I find that you asking people how they came up with the answer is rather pointless. If anyone writes out a rigorous proof that this is the largest, or even just a solution to arrive at this prime, THEN I will eat my own words. Daniel Liu · 3 years, 8 months ago

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@Daniel Liu People want to know if there is an elegant way to find the largest superprime. It may be possible that the solution requires the use of computers. This discussion is to clear that out. Bruce Wayne · 3 years, 8 months ago

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me too 73939133 Devansh Shringi · 3 years, 8 months ago

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@Devansh Shringi Yup.The largest superprime is 73939133. Can you show how you arrived at this answer? Bruce Wayne · 3 years, 8 months ago

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@Bruce Wayne Is there any way to prove that the largest superprime is 73939133? Mateo Torres · 3 years, 8 months ago

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73939133 Bobby Jim · 3 years, 8 months ago

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