## Left-truncatable Primes

Order 1
The largest left-truncatable prime is 24 digits: 357686312646216567629137.
Removing a digit at a time from the left end gives a sequence of 24 primes.
The prime has been found independently by several people.
There are 4260 left-truncatable primes.

Order 2
The 184-digit p184 is the largest known left-truncatable prime of order 2:

```p184 =                                                  3317\
676039361863813375186047305269232254334434985415346321657293\
347842184166316972712521507542402061477899339469603596634858\
212099979878129094817736602146359724182316273512181213141511```
This means that removing 2 digits at a time from the left end gives a sequence of 92 primes: p184, ..., 1511, 11.
p184 is also a left-truncatable base-100 prime.
The primes were certified by Marcel Martin's Primo (1511 and 11 were too small for Primo).

The prime was found by starting with 11 and recursively trying to add two digits from 10 to 99 on the left end. This gives 90 prime possibilities in each step.
The chance of a random n-digit number ending in 1, 3, 7 or 9 being prime, is around 1 in log(10^n) * 1/2 * 4/5 = 1 in 0.921n. From 98 to 184 digits this goes from 1 in 90 to 1 in 169. 98 digits was easy to reach but 43 further steps to 184 digits required a lot of tries (and went back to 28 digits at one time).

Order 3
The 1140-digit p1140 is the largest known left-truncatable prime of order 3:

```p1140 =
686957720511887558525174658414510840176432151923825981304948\
378237135960629558400414747213755738286767792781351488750517\
264488672424143774793266286770364857174294438649140883405465\
111650184405422520731603936126112798650690193956270492667782\
252202425468175747633902739432648860601275832729600906840482\
517851207730351684240852483171368592354596760617184207344771\
303768615441256104709615257106329207958857891370636668654226\
668627598741990159293937377616627380523797310848321354613345\
824936640519215403201542663630168337125631393644321198900751\
309897458118224375862155562138132204372567203408924412447426\
191762625165468828155675928128109122396184327504132254486363\
462573142336434187126453194249586173168628353985230306916320\
165307176186115255273138159294501491217530102244194102187144\
270207207114774123518132546106144140201166182339319824304125\
676185525801369789378639298750449400209313132470714271252396\
201303215168177114238260253260193102638141177189196377117229\
113172542190366267119229110112420351285173283143252142138105\
125265215598270147245157170100165362100128133129158132132180\
111172135104111124168102122154101132100107102106105101102101
```
This means that removing 3 digits at a time from the left end gives a sequence of 380 primes.
p1140 is also a left-truncatable base-1000 prime.

It was found in the same way as p184. This time there are 900 possibilities in each step, adding 100 to 999 on the left end. The primality chance goes from 1 in 901 at 978 digits to 1 in 1050 at 1140 digits (54 steps later). Fewer tries were needed than for p184, but each try took many times longer because of the often titanic size.

The 380 primes (except 101 which is too small) were certified by Primo in 99 cpu hours on an Athlon XP 1500+.

## Right-truncatable Primes

Order 1
The largest right-truncatable prime is 73939133.
Removing a digit at a time from the right end gives a sequence of 8 primes.
There are 83 right-truncatable primes.
There are much fewer right- than left-truncatable primes because all digits except the first must be 1, 3, 7 or 9 in a right-truncatable prime.
Longer sequences (probably arbitrarily long) become possible if it doesn't have to end with a 1-digit prime.
Example: 15243423901999999999 can be truncated 9 times (removing a 9 each time) to 15243423901.

Order 2
The only 110-digit right-truncatable prime of order 2 and starting with "11":

```112997419307834977573171270727470309575119399999236391538737\
53018739231353934953196323876313992301272907878337```
Removing 2 digits at a time from the right end gives a sequence of 55 primes. Only primes starting with "11" were searched.