Truncatable primes
You are encouraged to solve this task according to the task description, using any language you may know.
A truncatable prime is prime number that when you successively remove digits from one end of the prime, you are left with a new prime number; for example, the number 997 is called a left-truncatable prime as the numbers 997, 97, and 7 are all prime. The number 7393 is a right-truncatable prime as the numbers 7393, 739, 73, and 7 formed by removing digits from its right are also prime. No zeroes are allowed in truncatable primes.
The task is to find the largest left-truncatable and right-truncatable primes less than one million.
C.f: Sieve of Eratosthenes; Truncatable Prime from Mathworld.
Haskell
Using
from HackageDB
<lang haskell>import Data.Numbers.Primes(primes, isPrime) import Data.List import Control.Arrow
primes1e6 = reverse. filter (all (/='0'). show) $ takeWhile(<=1000000) primes
rightT, leftT :: Int -> Bool rightT = all isPrime. takeWhile(>0). drop 1. iterate (`div`10) leftT x = all isPrime. takeWhile(<x).map (x`mod`) $ iterate (*10) 10
main = do
let (ltp, rtp) = (head. filter leftT &&& head. filter rightT) primes1e6 putStrLn $ "Left truncatable " ++ show ltp putStrLn $ "Right truncatable " ++ show rtp</lang>
Output: <lang haskell>*Main> main Left truncatable 998443 Right truncatable 739399</lang>
J
Truncatable primes may be constructed by starting with a set of one digit prime numbers and then repeatedly adding a non-zero digit and selecting the prime numbers which result.
In other words, given:
<lang j>selPrime=: #~ 1&p: seed=: selPrime digits=: 1+i.9 step=: selPrime@,@:(,&.":/&>)@{@;</lang>
The largest truncatable primes less than a million can be obtained by adding five digits to the prime seeds, then finding the largest value from the result:
<lang j> >./ digits&step^:5 seed NB. left truncatable 998443
>./ step&digits^:5 seed NB. right truncatable
739399</lang>
Python
<lang python>maxprime = 1000000
def primes(n):
multiples = set()
prime = []
for i in range(2, n+1):
if i not in multiples:
prime.append(i)
multiples.update(set(range(i*i, n+1, i)))
return prime
def truncatableprime(n):
'Return a longest left and right truncatable primes below n'
primelist = [str(x) for x in primes(n)[::-1]]
primeset = set(primelist)
for n in primelist:
# n = 'abc'; [n[i:] for i in range(len(n))] -> ['abc', 'bc', 'c']
alltruncs = set(n[i:] for i in range(len(n)))
if alltruncs.issubset(primeset):
truncateleft = int(n)
break
for n in primelist:
# n = 'abc'; [n[:i+1] for i in range(len(n))] -> ['a', 'ab', 'abc']
alltruncs = set([n[:i+1] for i in range(len(n))])
if alltruncs.issubset(primeset):
truncateright = int(n)
break
return truncateleft, truncateright
print(truncatableprime(maxprime))</lang>
Sample Output
(998443, 739399)
Tcl
<lang tcl>package require Tcl 8.5
- Optimized version of the Sieve-of-Eratosthenes task solution
proc sieve n {
set primes [list]
if {$n < 2} {return $primes}
set nums [dict create]
for {set i 2} {$i <= $n} {incr i} {
dict set nums $i ""
}
set next 2
set limit [expr {sqrt($n)}]
while {$next <= $limit} {
for {set i $next} {$i <= $n} {incr i $next} {dict unset nums $i}
lappend primes $next
dict for {next -} $nums break
} return [concat $primes [dict keys $nums]]
}
proc isLeftTruncatable n {
global isPrime
while {[string length $n] > 0} {
if {![info exist isPrime($n)]} { return false } set n [string range $n 1 end]
} return true
} proc isRightTruncatable n {
global isPrime
while {[string length $n] > 0} {
if {![info exist isPrime($n)]} { return false } set n [string range $n 0 end-1]
} return true
}
- Demo code
set limit 1000000 puts "calculating primes up to $limit" set primes [sieve $limit] puts "search space contains [llength $primes] members" foreach p $primes {
set isPrime($p) "yes"
} set primes [lreverse $primes]
puts "searching for largest left-truncatable prime" foreach p $primes {
if {[isLeftTruncatable $p]} {
puts FOUND:$p break
}
}
puts "searching for largest right-truncatable prime" foreach p $primes {
if {[isRightTruncatable $p]} {
puts FOUND:$p break
}
}</lang> Output:
calculating primes up to 1000000 search space contains 78498 members searching for largest left-truncatable prime FOUND:998443 searching for largest right-truncatable prime FOUND:739399