Triangle Of The Gods
November 10, 2015
This is a trick question. No such prime number is known, though it is conjectured that an infinite number of them exist. All terms of the sequence up to n = 270,000 have been tested and found to be composite. Here’s our program to generate and test elements of the sequence:
> (let loop ((i 1) (n 1))
(display n) (newline)
(if (prime? n) (display "PRIME!")
(loop (+ i 1) (+ (* n (expt 10 (size (+ i 1)))) i 1))))
1
12
123
1234
...
You can run the program at http://ideone.com/hoYqVc, where you will also see the definitions of ilog, prime? and size.
Programming Praxis 
Ok, I’ll be the first one to admit I got trolled :P
Sequence generator:
Scala:
1. stream of Smarandache numbers
2. stream of primes
3. take te common elements
def sqrt(number: BigInt) = { val one = BigInt(1) def next(n: BigInt, i: BigInt): BigInt = (n + i / n) >> 1 var n = one var n1 = next(n, number) while ((n1 - n).abs > one) { n = n1 n1 = next(n, number) } while (n1 * n1 > number) { n1 -= one } n1 } //> sqrt: (number: BigInt)BigInt def smarandacheBI(n: BigInt = 1): Stream[BigInt] = { if (n % 2 == 0 || n % 5 == 0) smarandacheBI(n + 1) else { val akt = (for (s <- BigInt(1) to n) yield s.toString()).reduce(_ + _) if(akt.map(_.toInt).sum % 3==0) smarandacheBI(n + 1) else { println(akt) Stream.cons(BigInt(akt), smarandacheBI(n + BigInt(1))) } } } //> smarandacheBI: (n: BigInt)Stream[BigInt] def isPrimeBI(n: BigInt, d: BigInt = BigInt(2)): Boolean = { //println(n+" "+d) if (n == 1) false else if (d >= sqrt(n)) true else if (n % d == 0) false else isPrimeBI(n, d + 1) } //> isPrimeBI: (n: BigInt, d: BigInt)Boolean smarandacheBI().filter(isPrimeBI(_)).take(1).toListActually, I’ve had more fun factoring elements of the Smarandache sequence than searching for primes:
Funny that these days I think of writing a generator function first. I could’ve just included that logic in the loop. I blame all the playing around with ranges (i.e. lazy sequences) that I’ve been doing.
I reused the isprime function (not shown) that I’d written for an earlier collatz primes exercise.
#include <sstream> #include <string> #include <algorithm> #include <iterator> #include <iostream> #include <boost/multiprecision/cpp_int.hpp> #include "isprime.hpp" using Int = boost::multiprecision::cpp_int; auto triangle = [current = Int(1), oss = std::ostringstream()]() mutable { oss << current++; return Int(oss.str()); }; int main() { while (true) { Int candidate = triangle(); bool prime = isprime(candidate); std::cout << candidate << ": " << prime << '\n'; if (prime) break; } }