Problems & Puzzles: Puzzles

Puzzle 2.- Prime strings

All the following numbers are primes : 

(cutting by the right most digit) 

73939133 
7393913 
739391 
73939 
7393 
739 
73 
7 

(cutting by the left most digit) 

933739397 
33739397 
3739397 
739397 
39397 
9397 
397 
97 
7 

Isn’t it interesting that the both record strings grow from 7 ? Can you produce other type of interesting and larger sequences ? 

http://www.astro.virginia.edu/~eww6n/math/PrimeString.html 

These are other strings I produced (19/07/98) following a suggestion of Patrick De Geest: all of them are pal-primes using pal-primes of three digits 

                            383 
                     313383313  
              353313383313353 
       757353313383313353757 
383757353313383313353757383 

                            383 
                     313383313  
              353313383313353 
       757353313383313353757 
929757353313383313353757929 

                            919 
                     101919101 
              151101919101151          
       131151101919101151131 
181131151101919101151131181 

                            919 
                     101919101 
              151101919101151          
       131151101919101151131 
919131151101919101151131919 

Solution

Vas Danilov from Russia sent this string: 

357686312646216567629137

He didn`t tell us but it’s a "cutting by the left most digit" sequence and, by the way the largest possible of this type.

It seems that several people has found it before.

By example see the Caldwell Glossary entry for "Left-truncatable primes" (http://www.utm.edu/research/primes/glossary/LeftTruncatablePrime.html).

There you can see that: "The three largest left-truncatable primes are: 

959 18918 99765 33196 93967…….(23 digits)

966 86312 64621 65676 29137……..(23 digits)

3576 86312 64621 65676 29137……..(24 digits)

who is (are) the author(s) of this sequences?

...

Today (12/July/98) I have just obtained this other prime-string:

"The Largest Palprime formed linking 3 digits-palprimes being primes all
his sub-strings (cut by the left)"

797181191787191757191787101797 (30 digits)
181191787191757191787101797
191787191757191787101797
787191757191787101797
191757191787101797
757191787101797
191787101797
787101797
101797
797

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Luke Pebody wrote (June 2005):

31-digit truncatable primes

6279821572756282163864793777199
8833367216294578819534799139337
8939662423123592347173339993799

can be truncated to single digit by combination of left-truncation and right-truncation steps

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Wilfred Whiteside wrote on Dec. 2005:

I tried to beat the 31 digit left/right truncateable Luke Peabody results (3 of them) posted for puzzle 2.  My recursive program spit out his same 3 numbers and there are no bigger solutions.  He was probably aware that there are no larger numbers of this type.

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J. K. Andersen has come (January, 2006) with a generalization of this puzzle:

Let p184 = 331767603936186381337518604730526923225433443498541534\
63216572933478421841663169727125215075424020614778993394696035966\
34858212099979878129094817736602146359724182316273512181213141511

Removing 2 digits at a time from the left end of p184 gives a sequence of 92 primes.

Let p1140 =
686957720511887558525174658414510840176432151923825981304948\
378237135960629558400414747213755738286767792781351488750517\
264488672424143774793266286770364857174294438649140883405465\
111650184405422520731603936126112798650690193956270492667782\
252202425468175747633902739432648860601275832729600906840482\
517851207730351684240852483171368592354596760617184207344771\
303768615441256104709615257106329207958857891370636668654226\
668627598741990159293937377616627380523797310848321354613345\
824936640519215403201542663630168337125631393644321198900751\
309897458118224375862155562138132204372567203408924412447426\
191762625165468828155675928128109122396184327504132254486363\
462573142336434187126453194249586173168628353985230306916320\
165307176186115255273138159294501491217530102244194102187144\
270207207114774123518132546106144140201166182339319824304125\
676185525801369789378639298750449400209313132470714271252396\
201303215168177114238260253260193102638141177189196377117229\
113172542190366267119229110112420351285173283143252142138105\
125265215598270147245157170100165362100128133129158132132180\
111172135104111124168102122154101132100107102106105101102101

Removing 3 digits at a time from the left end of p1140 gives a sequence of 380 primes.

More details at http://hjem.get2net.dk/jka/math/left-truncatable.htm

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