# The largest Thursday number

A Thursday number is a number $N$ where any three consecutive digits make a prime, and all such primes formed are distinct.

For example, $13739$ is a Thursday number because $137$, $373$ and $739$ are all distinct primes.

Find the largest Thursday number. (Preferably without using a program)

P.S. I know today is not yet Thursday, but I may not be able to come online tomorrow.

• some people do not seem to like the number puzzles looking at the downvotes. Personally, I like them :) – Ivo Beckers Sep 30 '15 at 8:48
• Are leading zeroes allowed? for example: would 10113 be allowed because 101, 011 and 113 are prime? – Ivo Beckers Sep 30 '15 at 9:01
• @IvoBeckers Even if they are it will hardly be useful, since they can only be used in the first two digits. – Rohcana Sep 30 '15 at 9:52

I believe it is

9419919379773971911373313179

containing these primes

941-419-199-991-919-193-937-379-797-977-773-739-397-971-719-191-911-113-137-373-733-331-313-131-317-179

I wrote a computer program for this that brute forced it. Of course computer programs are prone to errors but I believe it's right.

• Sorry for the many edits. had some bugs in my code, but now I truly believe this is my final answer – Ivo Beckers Sep 30 '15 at 11:48
• I wrote a program to prove you wrong. Unfortunately, it only confirmed your answer. So +1. – Joel Rondeau Sep 30 '15 at 19:29

76199197739719373311313797

Work still in progress

Same as with the Wednesday number, we can only use $1$, $3$, $7$ and $9$ in primes, except for the first one, so we only have 30 primes that we can use:

113 131 137 139 173 179 191 193 197 199
311 313 317 331 337 373 379 397
719 733 739 773 797
911 919 937 971 977 991 997


Same as with the Wednesday number we should make a longest number with them. Observing the numbers gives us the following:

$$\begin{array}{lrr} &\text{Starts:}&\text{Ends:}\\ \hline 11&1&2\\ 13&3&2\\ 17&2&1\\ 19&4&2\\ 31&3&2\\ 33&2&1\\ 37&2&3\\ 39&1&2\\ 71&1&1\\ 73&2&3\\ 77&1&1\\ 79&1&2\\ 91&2&2\\ 93&1&1\\ 97&2&4\\ 99&2&1 \end{array}$$

Where column Starts means that a prime starts with that number, Ends means that the prime ends with that number and numbers are just # of occurrences. Finding a 1 - 1 pairs we can immediately say, that they come after each other:

71, 77, 93:
9719, 9773 and 1937

After that it's just a bit of playing around trying to find the longest number. It will consist out of 23 3-digit primes(taking the sum of the lowest of each row in the table), so 25 digits long.

• Well, you do need to check if the resulting number actually is prime... – Tim Couwelier Sep 30 '15 at 11:54
• @TimCouwelier that isn't a requirement – Ivo Beckers Sep 30 '15 at 11:56
• @TimCouwelier why should I? It doesn't say so in the question – Novarg Sep 30 '15 at 11:57
• @Novarg - It appears I got a bit over-ambitious. You're right. It would make this puzzle ALOT harder though. – Tim Couwelier Sep 30 '15 at 12:09