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 ?

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

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– Bruce Wayne · 3 years, 8 months ago

Yup.The largest superprime is 73939133. Can you show how you arrived at this answer?Log in to reply

– Akshay Ginodia · 3 years, 8 months ago

how did u get it?Log in to reply

– Patrick Corn · 3 years, 8 months ago

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.Log in to reply

– Mark Hennings · 3 years, 8 months ago

Well, \(L_1=\{2,3,5,7\}\), so you are missing a few. I think that \(L_8\) has about five elements.Log in to reply

– Patrick Corn · 3 years, 8 months ago

Yes indeed, thanks for pointing that out! I agree that \( |L_8| = 5 \).Log in to reply

– Gaurav Jain · 2 years, 5 months ago

elegant approachLog in to reply

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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– Shubham Kumar · 3 years, 8 months ago

Is their any solid reason or proof for the 'statement' you stated. If their then please provide.Log in to reply

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

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mustbe 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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– Michael Tong · 3 years, 8 months ago

The number of palindromic primes and the number of emirps (e.g. whether they're infinite or not) are both open problems.Log in to reply

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

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– Harvey Dent · 3 years, 8 months ago

Nops :P. I just use this name every now and then. I have seen screen names like 'Harvey Dent' and 'Batman' too.Log in to reply

– Ranjana Kasangeri · 3 years, 8 months ago

I don't think so!Log in to reply

Weeks ago I did this:

http://archives.somee.comto search for prime numbers, I think I could easily add a method to search forsuperprimes. But I'd love to see someone solve this by pure mathematics. – Mateo Torres · 3 years, 8 months agoLog in to reply

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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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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– Bruce Wayne · 3 years, 8 months ago

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.Log in to reply

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

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– Bruce Wayne · 3 years, 8 months ago

Yup.The largest superprime is 73939133. Can you show how you arrived at this answer?Log in to reply

– Mateo Torres · 3 years, 8 months ago

Is there any way to prove that the largest superprime is 73939133?Log in to reply

73939133 – Bobby Jim · 3 years, 8 months ago

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