Erdős-primes

In mathematics, Erdős primes are prime numbers such that all p-k! for 1<=k!<p are composite.

Erdős-primes is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
Definitions
Task

Write a program to determine (and show here) all Erdős primes less than 2500.

Optionally, show the number of Erdős primes.

Stretch goal

Show that the 7,875th Erdős prime is 999,721 (the highest below 1,000,000)

Also see



11lEdit

Translation of: Nim
F primes_upto(limit)
   V is_prime = [0B] * 2 [+] [1B] * (limit - 1)
   L(n) 0 .< Int(limit ^ 0.5 + 1.5)
      I is_prime[n]
         L(i) (n * n .. limit).step(n)
            is_prime[i] = 0B
   R is_prime

V is_prime = primes_upto(1'000'000)
V primeList = enumerate(is_prime).filter((i, prime) -> prime).map((i, prime) -> i)

[Int] factorials
L(n) 1..
   I factorial(n) >= 1'000'000
      L.break
   factorials.append(factorial(n))

F isErdosPrime(p)
   L(f) :factorials
      I f >= p
         L.break
      I :is_prime[p - f]
         R 0B
   R 1B

[Int] erdosList2500
L(p) primeList
   I p >= 2500
      L.break
   I isErdosPrime(p)
      erdosList2500.append(p)

print(‘Found ’erdosList2500.len‘ Erdos primes less than 2500:’)
L(prime) erdosList2500
   print(‘#5’.format(prime), end' I (L.index + 1) % 10 == 0 {"\n"} E ‘ ’)
print()

V count = 0
L(p) primeList
   I isErdosPrime(p)
      count++
      I count == 7875
         print("\nThe 7875th Erdos prime is "p‘.’)
         L.break
Output:
Found 25 Erdos primes less than 2500:
    2   101   211   367   409   419   461   557   673   709
  769   937   967  1009  1201  1259  1709  1831  1889  2141
 2221  2309  2351  2411  2437 

The 7875th Erdos prime is 999721.

Action!Edit

INCLUDE "H6:SIEVE.ACT"

BYTE Func IsErdosPrime(INT x BYTE ARRAY primes)
  INT k,f

  IF primes(x)=0 THEN
    RETURN (0)
  FI

  k=1 f=1
  WHILE f<x
  DO
    IF primes(x-f) THEN
      RETURN (0)
    FI
    k==+1
    f==*k
  OD
RETURN (1)

PROC Main()
  DEFINE MAX="2499"
  BYTE ARRAY primes(MAX+1)
  INT i,count=[0]

  Put(125) PutE() ;clear the screen
  Sieve(primes,MAX+1)
  FOR i=2 TO MAX
  DO
    IF IsErdosPrime(i,primes) THEN
      PrintI(i) Put(32)
      count==+1
    FI
  OD
  PrintF("%E%EThere are %I Erdos primes",count)
RETURN
Output:

Screenshot from Atari 8-bit computer

2 101 211 367 409 419 461 557 673 709 769 937 967 1009 1201 1259 1709 1831 1889 2141 2221 2309 2351 2411 2437

There are 25 Erdos primes

ALGOL 68Edit

BEGIN # find Erdős primes - primes p such that p-k! is composite for all 1<=k!<p #
    # returns TRUE if p is an Erdős prime #
    PROC is erdos prime = ( INT p )BOOL:
         IF NOT prime[ p ]
         THEN FALSE
         ELSE
            BOOL result := TRUE;
            FOR k WHILE factorial[ k ] < p AND result DO
                result := NOT prime[ p - factorial[ k ] ]
            OD;
            result
         FI # is erdos prime # ;
    INT max prime = 2500; # maximum number we will consider #
    # construct a table of factorials large enough for max prime #
    [ 1 : 12 ]INT factorial;
    factorial[ 1 ] := 1;
    FOR f FROM 2 TO UPB factorial DO
        factorial[ f ] := factorial[ f - 1 ] * f
    OD;
    # sieve the primes to max prime #
    PR read "primes.incl.a68" PR
    []BOOL prime = PRIMESIEVE max prime;
    # find the Erdős primes #
    INT e count := 0;
    IF is erdos prime( 2 ) THEN
        print( ( " 2" ) );
        e count +:= 1
    FI;
    FOR p FROM 3 BY 2 TO UPB prime DO
        IF is erdos prime( p ) THEN
            print( ( " ", whole( p, 0 ) ) );
            e count +:= 1
        FI
    OD;
    print( ( newline, "Found ", whole( e count, 0 ), " Erdos primes" ) )
END
Output:
 2 101 211 367 409 419 461 557 673 709 769 937 967 1009 1201 1259 1709 1831 1889 2141 2221 2309 2351 2411 2437
Found 25 Erdos primes

ArturoEdit

factorials: map 1..20 => [product 1..&]
erdos?: function [x][
    if not? prime? x -> return false

    loop factorials 'f [
        if f >= x -> break
        if prime? x - f -> return false
    ]
    return true
]

loop split.every:10 select 2..2500 => erdos? 'a ->
    print map a => [pad to :string & 5]
Output:
    2   101   211   367   409   419   461   557   673   709 
  769   937   967  1009  1201  1259  1709  1831  1889  2141 
 2221  2309  2351  2411  2437

AWKEdit

# syntax: GAWK -f ERDOS-PRIMES.AWK
# converted from FreeBASIC
BEGIN {
    while (++i) {
      if (is_erdos_prime(i)) {
        if (i < 2500) {
          printf("%d ",i)
          count1++
        }
        if (++count2 == 7875) {
          printf("\nErdos primes 1-2500: %d\nErdos prime %d: %d\n",count1,count2,i)
          break
        }
      }
    }
    exit(0)
}
function is_erdos_prime(p,  kf,m) {
    if (!is_prime(p)) { return(0) }
    kf = m = 1
    while (kf < p) {
      kf *= m++
      if (is_prime(p-kf)) { return(0) }
    }
    return(1)
}
function is_prime(x,  i) {
    if (x <= 1) {
      return(0)
    }
    for (i=2; i<=int(sqrt(x)); i++) {
      if (x % i == 0) {
        return(0)
      }
    }
    return(1)
}
Output:
2 101 211 367 409 419 461 557 673 709 769 937 967 1009 1201 1259 1709 1831 1889 2141 2221 2309 2351 2411 2437
Erdos primes 1-2500: 25
Erdos prime 7875: 999721

C#Edit

using System; using static System.Console;
class Program {
  const int lmt = (int)1e6, first = 2500; static int[] f = new int[10];
  static void Main(string[] args) {
    f[0] = 1; for (int a = 0, b = 1; b < f.Length; a = b++)
      f[b] = f[a] * (b + 1);
    int pc = 0, nth = 0, lv = 0;
    for (int i = 2; i < lmt; i++) if (is_erdos_prime(i)) {
        if (i < first) Write("{0,5:n0}{1}", i, pc++ % 5 == 4 ? "\n" : "  ");
        nth++; lv = i; }
    Write("\nCount of Erdős primes between 1 and {0:n0}: {1}\n{2} Erdős prime (the last one under {3:n0}): {4:n0}", first, pc, ord(nth), lmt, lv); }

  static string ord(int n) {
    return string.Format("{0:n0}", n) + new string[]{"th", "st", "nd", "rd", "th", "th", "th", "th", "th", "th"}[n % 10]; }

  static bool is_erdos_prime(int p) {
    if (!is_pr(p)) return false; int m = 0, t;
    while ((t = p - f[m++]) > 0) if (is_pr(t)) return false;
    return true;
    bool is_pr(int x) {
      if (x < 4) return x > 1; if ((x & 1) == 0) return false;
      for (int i = 3; i * i <= x; i += 2) if (x % i == 0) return false;
    return true; } } }
Output:
    2    101    211    367    409
  419    461    557    673    709
  769    937    967  1,009  1,201
1,259  1,709  1,831  1,889  2,141
2,221  2,309  2,351  2,411  2,437

Count of Erdős primes between 1 and 2,500: 25
7,875th Erdős prime (the last one under 1,000,000): 999,721

C++Edit

Library: Primesieve
#include <cstdint>
#include <iomanip>
#include <iostream>
#include <set>
#include <primesieve.hpp>

class erdos_prime_generator {
public:
    erdos_prime_generator() {}
    uint64_t next();
private:
    bool erdos(uint64_t p) const;
    primesieve::iterator iter_;
    std::set<uint64_t> primes_;
};

uint64_t erdos_prime_generator::next() {
    uint64_t prime;
    for (;;) {
        prime = iter_.next_prime();
        primes_.insert(prime);
        if (erdos(prime))
            break;
    }
    return prime;
}

bool erdos_prime_generator::erdos(uint64_t p) const {
    for (uint64_t k = 1, f = 1; f < p; ++k, f *= k) {
        if (primes_.find(p - f) != primes_.end())
            return false;
    }
    return true;
}

int main() {
    std::wcout.imbue(std::locale(""));
    erdos_prime_generator epgen;
    const int max_print = 2500;
    const int max_count = 7875;
    uint64_t p;
    std::wcout << L"Erd\x151s primes less than " << max_print << L":\n";
    for (int count = 1; count <= max_count; ++count) {
        p = epgen.next();
        if (p < max_print)
            std::wcout << std::setw(6) << p << (count % 10 == 0 ? '\n' : ' ');
    }
    std::wcout << L"\n\nThe " << max_count << L"th Erd\x151s prime is " << p << L".\n";
    return 0;
}
Output:
Erdős primes less than 2,500:
     2    101    211    367    409    419    461    557    673    709
   769    937    967  1,009  1,201  1,259  1,709  1,831  1,889  2,141
 2,221  2,309  2,351  2,411  2,437 

The 7,875th Erdős prime is 999,721.

F#Edit

This task uses Extensible Prime Generator (F#)

// Erdős Primes. Nigel Galloway: March 22nd., 2021
let rec fN g=function 1->g |n->fN(g*n)(n-1)
let rec fG n g=seq{let i=fN 1 n in if i<g then yield (isPrime>>not)(g-i); yield! fG(n+1) g}
let eP()=primes32()|>Seq.filter(fG 1>>Seq.forall id)
eP()|>Seq.takeWhile((>)2500)|>Seq.iter(printf "%d "); printfn "\n\n7875th Erdős prime is %d" (eP()|>Seq.item 7874)
Output:
2 101 211 367 409 419 461 557 673 709 769 937 967 1009 1201 1259 1709 1831 1889 2141 2221 2309 2351 2411 2437

7875th Erdos prime is 999721

FactorEdit

Works with: Factor version 0.99 2021-02-05
USING: formatting io kernel lists lists.lazy math
math.factorials math.primes math.primes.lists math.vectors
prettyprint sequences tools.memory.private ;

: facts ( -- list ) 1 lfrom [ n! ] lmap-lazy ;

: n!< ( p -- seq ) [ facts ] dip [ < ] curry lwhile list>array ;

: erdős? ( p -- ? ) dup n!< n-v [ prime? ] none? ;

: erdős ( -- list ) lprimes [ erdős? ] lfilter ;

erdős [ 2500 < ] lwhile list>array
dup length "Found  %d  Erdős primes < 2500:\n" printf
[ bl ] [ pprint ] interleave nl nl

7874 erdős lnth commas
"The 7,875th Erdős prime is %s.\n" printf
Output:
Found  25  Erdős primes < 2500:
2 101 211 367 409 419 461 557 673 709 769 937 967 1009 1201 1259 1709 1831 1889 2141 2221 2309 2351 2411 2437

The 7,875th Erdős prime is 999,721.

ForthEdit

Works with: Gforth
: prime? ( n -- ? ) here + c@ 0= ;
: notprime! ( n -- ) here + 1 swap c! ;

: prime_sieve { n -- }
  here n erase
  0 notprime!
  1 notprime!
  n 4 > if
    n 4 do i notprime! 2 +loop
  then
  3
  begin
    dup dup * n <
  while
    dup prime? if
      n over dup * do
        i notprime!
      dup 2* +loop
    then
    2 +
  repeat
  drop ;

: erdos_prime? { p -- ? }
  p prime? if
    1 1
    begin
      dup p <
    while
      p over - prime? if 2drop false exit then
      swap 1+ swap
      over *
    repeat
    2drop true
  else
    false
  then ;  

: print_erdos_primes { n -- }
  ." Erdos primes < " n 1 .r ." :" cr
  n prime_sieve
  0
  n 0 do
    i erdos_prime? if
      i 5 .r
      1+ dup 10 mod 0= if cr then
    then
  loop
  cr ." Count: " . cr ;

2500 print_erdos_primes
bye
Output:
Erdos primes < 2500:
    2  101  211  367  409  419  461  557  673  709
  769  937  967 1009 1201 1259 1709 1831 1889 2141
 2221 2309 2351 2411 2437
Count: 25 

FreeBASICEdit

I won't bother reproducing a primality-testing function; use the one from Primality_by_trial_division#FreeBASIC.

#include "isprime.bas"

function is_erdos_prime( p as uinteger ) as boolean
    if not isprime(p) then return false
    dim as uinteger kf=1, m=1
    while kf < p
        kf*=m : m+=1
        if isprime(p - kf) then return false
    wend
    return true
end function

dim as integer c = 0, i = 1
while c<7875
    i+=1
    if is_erdos_prime(i) then 
        c+=1
        if i<2500 or c=7875 then print c, i
    end if
wend
Output:
1             2

2 101 3 211 4 367 5 409 6 419 7 461 8 557 9 673 10 709 11 769 12 937 13 967 14 1009 15 1201 16 1259 17 1709 18 1831 19 1889 20 2141 21 2221 22 2309 23 2351 24 2411 25 2437 7875 999721


GoEdit

Translation of: Wren
package main

import "fmt"

func sieve(limit int) []bool {
    limit++
    // True denotes composite, false denotes prime.
    c := make([]bool, limit) // all false by default
    c[0] = true
    c[1] = true
    for i := 4; i < limit; i += 2 {
        c[i] = true
    }
    p := 3 // Start from 3.
    for {
        p2 := p * p
        if p2 >= limit {
            break
        }
        for i := p2; i < limit; i += 2 * p {
            c[i] = true
        }
        for {
            p += 2
            if !c[p] {
                break
            }
        }
    }
    return c
}

func commatize(n int) string {
    s := fmt.Sprintf("%d", n)
    if n < 0 {
        s = s[1:]
    }
    le := len(s)
    for i := le - 3; i >= 1; i -= 3 {
        s = s[0:i] + "," + s[i:]
    }
    if n >= 0 {
        return s
    }
    return "-" + s
}

func main() {
    limit := int(1e6)
    c := sieve(limit - 1)
    var erdos []int
    for i := 2; i < limit; {
        if !c[i] {
            found := true
            for j, fact := 1, 1; fact < i; {
                if !c[i-fact] {
                    found = false
                    break
                }
                j++
                fact = fact * j
            }
            if found {
                erdos = append(erdos, i)
            }
        }
        if i > 2 {
            i += 2
        } else {
            i += 1
        }
    }
    lowerLimit := 2500
    var erdosLower []int
    for _, e := range erdos {
        if e < lowerLimit {
            erdosLower = append(erdosLower, e)
        } else {
            break
        }
    }
    fmt.Printf("The %d Erdős primes under %s are\n", len(erdosLower), commatize(lowerLimit))
    for i, e := range erdosLower {
        fmt.Printf("%6d", e)
        if (i+1)%10 == 0 {
            fmt.Println()
        }
    }
    show := 7875
    fmt.Printf("\n\nThe %s Erdős prime is %s.\n", commatize(show), commatize(erdos[show-1]))
}
Output:
The 25 Erdős primes under 2,500 are
     2   101   211   367   409   419   461   557   673   709
   769   937   967  1009  1201  1259  1709  1831  1889  2141
  2221  2309  2351  2411  2437

The 7,875 Erdős prime is 999,721.

JEdit

Implementation:

NB. erdos primes greater than !k and less than or equal to !k+1 (where !k is the factorial of k)
erdospr=: {{ k=. y
  f=. !k+0 1
  p=. (#~ 1= f&I.) p:(+i.)/0 1+p:inv f
  p#~*/|:0=1 p:p-/!i.1+k
}}

NB. erdos primes less than j
erdosprs=: {{ (#~ j&>);erdospr&.>i.>.!inv j=. y }}

Task examples:

   erdosprs 2500
2 101 211 367 409 419 461 557 673 709 769 937 967 1009 1201 1259 1709 1831 1889 2141 2221 2309 2351 2411 2437
   (#,{:) erdosprs 1e6
7875 999721

JavaEdit

import java.util.*;

public class ErdosPrimes {
    public static void main(String[] args) {
        boolean[] sieve = primeSieve(1000000);
        int maxPrint = 2500;
        int maxCount = 7875;
        System.out.printf("Erd\u0151s primes less than %d:\n", maxPrint);
        for (int count = 0, prime = 1; count < maxCount; ++prime) {
            if (erdos(sieve, prime)) {
                ++count;
                if (prime < maxPrint) {
                    System.out.printf("%6d", prime);
                    if (count % 10 == 0)
                        System.out.println();
                }
                if (count == maxCount)
                    System.out.printf("\n\nThe %dth Erd\u0151s prime is %d.\n", maxCount, prime);
            }
        }
    }

    private static boolean erdos(boolean[] sieve, int p) {
        if (!sieve[p])
            return false;
        for (int k = 1, f = 1; f < p; ++k, f *= k) {
            if (sieve[p - f])
                return false;
        }
        return true;
    }

    private static boolean[] primeSieve(int limit) {
        boolean[] sieve = new boolean[limit];
        Arrays.fill(sieve, true);
        if (limit > 0)
            sieve[0] = false;
        if (limit > 1)
            sieve[1] = false;
        for (int i = 4; i < limit; i += 2)
            sieve[i] = false;
        for (int p = 3; ; p += 2) {
            int q = p * p;
            if (q >= limit)
                break;
            if (sieve[p]) {
                int inc = 2 * p;
                for (; q < limit; q += inc)
                    sieve[q] = false;
            }
        }
        return sieve;
    }
}
Output:
Erdős primes less than 2500:
     2   101   211   367   409   419   461   557   673   709
   769   937   967  1009  1201  1259  1709  1831  1889  2141
  2221  2309  2351  2411  2437

The 7875th Erdős prime is 999721.

jqEdit

Works with: jq

Works with gojq, the Go implementation of jq (but the second task requires an unreasonable amount of memory)


Preliminaries

def emit_until(cond; stream): label $out | stream | if cond then break $out else . end;

def is_prime:
  . as $n
  | if ($n < 2)         then false
    elif ($n % 2 == 0)  then $n == 2
    elif ($n % 3 == 0)  then $n == 3
    elif ($n % 5 == 0)  then $n == 5
    elif ($n % 7 == 0)  then $n == 7
    elif ($n % 11 == 0) then $n == 11
    elif ($n % 13 == 0) then $n == 13
    elif ($n % 17 == 0) then $n == 17
    elif ($n % 19 == 0) then $n == 19
    else 23
    | until( (. * .) > $n or ($n % . == 0); . + 2)
    | . * . > $n
    end;

Erdős-primes

 
def is_Erdos:
  . as $p
  | if is_prime|not then false
    else label $out
    | foreach range(1; .+1) as $k (1; . * $k;
        if . >= $p then true, break $out
        elif ($p - .) | is_prime then 0, break $out
	else empty
	end) // true
    | . == true
    end ;	

# emit the Erdos primes
def Erdos: range(2; infinite) | select(is_Erdos);

The tasks

"The Erdős primes less than 2500 are:", emit_until(. >= 2500; Erdos),

"\nThe 7875th Erdős prime is \(nth(7874; Erdos))."
Output:
The Erdős primes less than 2500 are:
2
101
211
367
409
419
461
557
673
709
769
937
967
1009
1201
1259
1709
1831
1889
2141
2221
2309
2351
2411
2437

The 7875th Erdős prime is 999721.

JuliaEdit

using Primes, Formatting

function isErdős(p::Integer)
    isprime(p) || return false
    for i in 1:100
        kfac = factorial(i)
        kfac >= p && break
        isprime(p - kfac) && return false
    end
    return true
end

const Erdőslist = filter(isErdős, 1:1000000)
const E2500 = filter(x -> x < 2500, Erdőslist)

println(length(E2500), " Erdős primes < 2500: ", E2500)
println("The 7875th Erdős prime is ", format(Erdőslist[7875], commas=true))
Output:
25 Erdős primes < 2500: [2, 101, 211, 367, 409, 419, 461, 557, 673, 709, 769, 937, 967, 1009, 1201, 1259, 1709, 1831, 1889, 2141, 2221, 2309, 2351, 2411, 2437]
The 7875th Erdős prime is 999,721

Mathematica/Wolfram LanguageEdit

ClearAll[ErdosPrimeQ]
ErdosPrimeQ[p_Integer] := Module[{k},
  If[PrimeQ[p],
   k = 1;
   While[k! < p,
    If[PrimeQ[p - k!], Return[False]];
    k++;
    ];
   True
   ,
   False
   ]
  ]
sel = Select[Range[2500], ErdosPrimeQ]
Length[sel]
sel = Select[Range[999999], ErdosPrimeQ];
{Length[sel], Last[sel]}
Output:
{2, 101, 211, 367, 409, 419, 461, 557, 673, 709, 769, 937, 967, 1009, 1201, 1259, 1709, 1831, 1889, 2141, 2221, 2309, 2351, 2411, 2437}
25
{7875, 999721}

NimEdit

import math, sets, strutils, sugar

const N = 1_000_000

# Sieve of Erathostenes.
var isComposite: array[2..N, bool]
for n in 2..N:
  let n2 = n * n
  if n2 > N: break
  if not isComposite[n]:
    for k in countup(n2, N, n):
      isComposite[k] = true

template isPrime(n: int): bool = n > 1 and not isComposite[n]

let primeList = collect(newSeq):
                  for n in 2..N:
                    if n.isPrime: n

const Factorials = collect(newSeq):
                     for n in 1..20:
                       if fac(n) >= N: break
                       fac(n)


proc isErdösPrime(p: int): bool =
  ## Check if prime "p" is an Erdös prime.
  for f in Factorials:
    if f >= p: break
    if (p - f).isPrime: return false
  result = true


let erdösList2500 = collect(newSeq):
                      for p in primeList:
                        if p >= 2500: break
                        if p.isErdösPrime: p

echo "Found $# Erdös primes less than 2500:".format(erdösList2500.len)
for i, prime in erdösList2500:
  stdout.write ($prime).align(5)
  stdout.write if (i+1) mod 10 == 0: '\n' else: ' '
echo()

var erdös7875: int
var count = 0
for p in primeList:
  if p.isErdösPrime: inc count
  if count == 7875:
    erdös7875 = p
    break
echo "\nThe 7875th Erdös prime is $#.".format(erdös7875)
Output:
Found 25 Erdös primes less than 2500:
    2   101   211   367   409   419   461   557   673   709
  769   937   967  1009  1201  1259  1709  1831  1889  2141
 2221  2309  2351  2411  2437 

The 7875th Erdös prime is 999721.

PerlEdit

Library: ntheory
use strict;
use warnings;
use feature 'say';
use utf8;
binmode(STDOUT, ':utf8');
use List::AllUtils 'before';
use ntheory qw<is_prime factorial>;

sub is_erdos {
    my($n) = @_; my $k;
    return unless is_prime($n);
    while ($n > factorial($k++)) { return if is_prime $n-factorial($k) }
    'True'
}

my(@Erdős,$n);
do { push @Erdős, $n if is_erdos(++$n) } until $n >= 1e6;

say 'Erdős primes < ' . (my $max = 2500) . ':';
say join ' ', before { $_ > 2500 } @Erdős;
say "\nErdős prime #" . @Erdős . ' is ' . $Erdős[-1];
Output:
Erdős primes < 2500:
2 101 211 367 409 419 461 557 673 709 769 937 967 1009 1201 1259 1709 1831 1889 2141 2221 2309 2351 2411 2437

Erdős prime #7875 is 999721

PhixEdit

atom t0 = time()
sequence facts = {1}
function erdos(integer p)
    while facts[$]<p do
        facts &= facts[$]*(length(facts)+1)
    end while
    for i=length(facts) to 1 by -1 do
        integer pmk = p-facts[i]
        if pmk>0 then
            if is_prime(pmk) then return false end if
        end if
    end for
    return true
end function
sequence res = filter(get_primes_le(2500),erdos)
printf(1,"Found %d Erdos primes < 2,500:\n%s\n\n",{length(res),join(apply(res,sprint))})
res = filter(get_primes_le(1000000),erdos)
integer l = length(res)
printf(1,"The %,d%s Erdos prime is %,d (%s)\n",{l,ord(l),res[$],elapsed(time()-t0)})
Output:
Found 25 Erdos primes < 2,500:
2 101 211 367 409 419 461 557 673 709 769 937 967 1009 1201 1259 1709 1831 1889 2141 2221 2309 2351 2411 2437

The 7,875th Erdos prime is 999,721 (1.2s)

RakuEdit

use Lingua::EN::Numbers;

my @factorial = 1, |[\*] 1..*;
my @Erdős = ^Inf .grep: { .is-prime and none($_ «-« @factorial[^(@factorial.first: * > $_, :k)]).is-prime }

put 'Erdős primes < 2500:';
put @Erdős[^(@Erdős.first: * > 2500, :k)]».&comma;
put "\nThe 7,875th Erdős prime is: " ~ @Erdős[7874].&comma;
Output:
Erdős primes < 2500:
2 101 211 367 409 419 461 557 673 709 769 937 967 1,009 1,201 1,259 1,709 1,831 1,889 2,141 2,221 2,309 2,351 2,411 2,437

The 7,875th Erdős prime is: 999,721

REXXEdit

/*REXX program counts/displays the number of  Erdos primes  under a specified number N. */
parse arg n cols .                               /*get optional number of primes to find*/
if    n=='' |    n==","  then    n= 2500         /*Not specified?   Then assume default.*/
if cols=='' | cols==","  then cols=   10         /* "      "          "     "       "   */
nn= n;                           n= abs(n)       /*N<0:  shows highest Erdos prime< │N│ */
call genP  n                                     /*generate all primes under  N.        */
w= 10                                            /*width of a number in any column.     */
if cols>0  then say ' index │'center(" Erdos primes that are  < "  n, 1 + cols*(w+1)     )
if cols>0  then say '───────┼'center(""                             , 1 + cols*(w+1), '─')
call facts                                       /*generate a table of needed factorials*/
Eprimes= 0;                     idx= 1           /*initialize # of additive primes & idx*/
$=                                               /*a list of additive primes  (so far). */
     do j=1  for #;             prime= @.j       /*                                     */
        do k=1  until fact.k>j                   /*verify: J-K! for 1≤K!<J are composite*/
        z= prime - fact.k                        /*subtract some factorial from a prime.*/
        if !.z  then iterate j                   /*Is   Z   is a prime?   Then skip it. */
        end   /*j*/

     Eprimes= Eprimes + 1;      EprimeL= j       /*bump the count of Erdos primes.      */
     if cols<0             then iterate          /*Build the list  (to be shown later)? */
     c= commas(j)                                /*maybe add some commas to the number. */
     $= $ right(c, max(w, length(c) ) )          /*add Erdos prime to list, allow big #.*/
     if Eprimes//cols\==0  then iterate          /*have we populated a line of output?  */
     say center(idx, 7)'│'  substr($, 2);   $=   /*display what we have so far  (cols). */
     idx= idx + cols                             /*bump the  index  count for the output*/
     end   /*j*/

if $\==''  then say center(idx, 7)"│"  substr($, 2)  /*possible display residual output.*/
if cols>0  then say '───────┴'center(""                             , 1 + cols*(w+1), '─')
say
say 'found '      commas(Eprimes)   " Erdos primes  < "                    commas(n)
say
if nn<0  then say commas(EprimeL)  ' is the '  commas(Eprimes)th(Eprimes)  " Erdos prime."
exit 0                                           /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg ?;  do jc=length(?)-3  to 1  by -3; ?=insert(',', ?, jc); end;  return ?
facts:  arg x; fact.=1;  do x=2 until fact.x>1e9; p= x-1; fact.x= x*fact.p; end;  return
th:     parse arg th; return word('th st nd rd', 1+(th//10) *(th//100%10\==1) *(th//10<4))
/*──────────────────────────────────────────────────────────────────────────────────────*/
genP: parse arg n;   @.=.; @.1=2;  @.2=3;  @.3=5;  @.4=7;  @.5=11;  @.6=13;  @.7=17;  #= 7
      w= length(n);  !.=0; !.2=1;  !.3=1;  !.5=1;  !.7=1;  !.11=1;  !.13=1;  !.17=1
            do j=@.7+2  by 2  while j<n          /*continue on with the next odd prime. */
            parse var  j  ''  -1  _              /*obtain the last digit of the  J  var.*/
            if _      ==5  then iterate          /*is this integer a multiple of five?  */
            if j // 3 ==0  then iterate          /* "   "     "    "     "     " three? */
                                                 /* [↓]  divide by the primes.   ___    */
                  do k=4  to #  while  k*k<=j    /*divide  J  by other primes ≤ √ J     */
                  if j//@.k == 0  then iterate j /*÷ by prev. prime?  ¬prime     ___    */
                  end   /*k*/                    /* [↑]   only divide up to     √ J     */
            #= # + 1;          @.#= j;  !.j= 1   /*bump prime count; assign prime & flag*/
            end   /*j*/;                return
output   when using the default inputs:
 index │                                         Erdos primes that are  <  2500
───────┼───────────────────────────────────────────────────────────────────────────────────────────────────────────────
   1   │          2        101        211        367        409        419        461        557        673        709
  11   │        769        937        967      1,009      1,201      1,259      1,709      1,831      1,889      2,141
  21   │      2,221      2,309      2,351      2,411      2,437
───────┴───────────────────────────────────────────────────────────────────────────────────────────────────────────────

found  25  Erdos primes  <  2500
output   when using the inputs of:     1000000   0
found  7,875  Erdos primes  <  1,000,000

999,721  is the  7,875th  Erdos prime.

RustEdit

// [dependencies]
// primal = "0.3"

use std::collections::HashSet;

fn erdos_primes() -> impl std::iter::Iterator<Item = usize> {
    let mut primes = HashSet::new();
    let mut all_primes = primal::Primes::all();
    std::iter::from_fn(move || {
        'all_primes: for p in all_primes.by_ref() {
            primes.insert(p);
            let mut k = 1;
            let mut f = 1;
            while f < p {
                if primes.contains(&(p - f)) {
                    continue 'all_primes;
                }
                k += 1;
                f *= k;
            }
            return Some(p);
        }
        None
    })
}

fn main() {
    let mut count = 0;
    println!("Erd\u{151}s primes less than 2500:");
    for p in erdos_primes().take_while(|x| *x < 2500) {
        count += 1;
        if count % 10 == 0 {
            println!("{:4}", p);
        } else {
            print!("{:4} ", p);
        }
    }
    println!();
    if let Some(p) = erdos_primes().nth(7874) {
        println!("\nThe 7875th Erd\u{151}s prime is {}.", p);
    }
}
Output:
Erdős primes less than 2500:
   2  101  211  367  409  419  461  557  673  709
 769  937  967 1009 1201 1259 1709 1831 1889 2141
2221 2309 2351 2411 2437 

The 7875th Erdős prime is 999721.

SidefEdit

func is_erdos_prime(p) {

    return true  if p==2
    return false if !p.is_prime

    var f = 1

    for (var k = 2; f < p; k++) {
        p - f -> is_composite || return false
        f *= k
    }

    return true
}

say ("Erdős primes <= 2500: ", 1..2500 -> grep(is_erdos_prime))
say ("The 7875th Erdős prime is: ", is_erdos_prime.nth(7875))
Output:
Erdős primes <= 2500: [2, 101, 211, 367, 409, 419, 461, 557, 673, 709, 769, 937, 967, 1009, 1201, 1259, 1709, 1831, 1889, 2141, 2221, 2309, 2351, 2411, 2437]
The 7875th Erdős prime is: 999721

Tiny BASICEdit

Can't manage the stretch goal because integers are limited to signed 16 bit.

     LET P = 1
  10 IF P > 2 THEN LET P = P + 2
     IF P < 3 THEN LET P = P + 1
     LET Z = P
     GOSUB 1000
     IF A = 0 THEN GOTO 10
     LET K = 0
  20 LET K = K + 1
     GOSUB 2000
     LET Z = P - F
     IF Z < 0 THEN GOTO 30
     GOSUB 1000
     IF A = 1 THEN LET E = 0
     IF A = 1 THEN GOTO 10
     GOTO 20
  30 LET C = C + 1
     IF P < 2500 THEN PRINT C,"  ",P
     IF P > 2500 THEN END
     GOTO 10
    
1000 REM primality of Z by trial division, result is in A
     LET Y = 1
     LET A = 0
     IF Z = 2 THEN LET A = 1
     IF Z < 3 THEN RETURN
1010 LET Y = Y + 2
     IF (Z/Y)*Y = Z THEN RETURN
     IF Y*Y < Z THEN GOTO 1010
     LET A = 1
     RETURN
     
2000 REM factorial of K, result is in F
     LET A = 1
     LET F = 1
2010 LET F = F*A
     IF A=K THEN RETURN
     LET A = A + 1
     GOTO 2010
Output:
1  2

2 101 3 211 4 367 5 409 6 419 7 461 8 673 9 709 10 769 11 937 12 967 13 1009 14 1201 15 1259 16 1709 17 1831 18 2141 19 2221 20 2351 21 2411 22 2437

WrenEdit

Library: Wren-math
Library: Wren-seq
Library: Wren-fmt
import "/math" for Int
import "/seq" for Lst
import "/fmt" for Fmt

var limit = 1e6
var primes = Int.primeSieve(limit - 1, true)
var erdos = []
for (p in primes) {
    var i = 1
    var fact = 1
    var found = true
    while (fact < p) {
        if (Int.isPrime(p - fact)) {
            found = false
            break
        }
        i = i + 1
        fact = fact * i
    }
    if (found) erdos.add(p)
}
var lowerLimit = 2500
var erdosLower = erdos.where { |e| e < lowerLimit}.toList
Fmt.print("The $,d Erdős primes under $,d are:", erdosLower.count, lowerLimit)
for (chunk in Lst.chunks(erdosLower, 10)) Fmt.print("$6d", chunk)
var show = 7875
Fmt.print("\nThe $,r Erdős prime is $,d.", show, erdos[show-1])
Output:
The 25 Erdős primes under 2,500 are:
     2    101    211    367    409    419    461    557    673    709
   769    937    967   1009   1201   1259   1709   1831   1889   2141
  2221   2309   2351   2411   2437

The 7,875th Erdős prime is 999,721.

XPL0Edit

func IsPrime(N);        \Return 'true' if N is prime
int  N, I;
[if N <= 2 then return N = 2;
if (N&1) = 0 then \even >2\ return false;
for I:= 3 to sqrt(N) do
    [if rem(N/I) = 0 then return false;
    I:= I+1;
    ];
return true;
];

func Erdos(N);          \Return 'true' if N is an Erdos prime
int  N, F, K;
[if not IsPrime(N) then return false;
F:= 1;  K:= 1;
while F < N do
    [if IsPrime(N-F) then return false;
    K:= K+1;  F:= F*K;
    ];
return true;
];

int Cnt, N, SN;
[Format(5, 0);
Cnt:= 0;  N:= 2;
repeat  if Erdos(N) then
            [RlOut(0, float(N));
            Cnt:= Cnt+1;
            if rem(Cnt/10) = 0 then CrLf(0);
            ];
        N:= N+1;
until   N >= 2500;
CrLf(0);
Text(0, "Found ");  IntOut(0, Cnt);  Text(0, " Erdos primes^m^j");
Cnt:= 1;  N:= 3;
repeat  if Erdos(N) then [Cnt:= Cnt+1;  SN:= N];
        N:= N+2;
until   N >= 1_000_000;
Text(0, "The ");  IntOut(0, Cnt);
Text(0, "th Erdos prime is indeed ");  IntOut(0, SN);  CrLf(0);
]
Output:
    2  101  211  367  409  419  461  557  673  709
  769  937  967 1009 1201 1259 1709 1831 1889 2141
 2221 2309 2351 2411 2437
Found 25 Erdos primes
The 7875th Erdos prime is indeed 999721