# Erdős-primes

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

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

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

## 11l

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.
```

## ABC

```HOW TO REPORT prime n:
SELECT:
n < 2: FAIL
n mod 2 = 0: REPORT n=2
ELSE: REPORT NO d IN {2..floor (root n)} HAS n mod d = 0

HOW TO REPORT erdos p:
IF NOT prime p: FAIL
PUT 1, 1 IN k, k.fac
WHILE k.fac < p:
IF prime (p - k.fac): FAIL
PUT k+1 IN k
PUT k.fac*k IN k.fac
SUCCEED

PUT 0 IN nprimes
FOR n IN {1..2499}:
IF erdos n:
WRITE n>>6
PUT nprimes+1 IN nprimes
IF nprimes mod 10 = 0: WRITE/

WRITE /
WRITE "There are `nprimes` Erdos primes < 2500."/

PUT 2499 IN n
WHILE nprimes < 7875:
PUT n+2 IN n
IF erdos n: PUT nprimes + 1 IN nprimes

WRITE "The `nprimes`th Erdos prime is `n`."/```
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
There are 25 Erdos primes < 2500.
The 7875th Erdos prime is 999721.```

## Action!

```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:
```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 68

```BEGIN # find Erdős primes - primes p where 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 = 1 000 000;             # maximum number we will consider #
INT max erdos =     7 875;                 # maximum Erdős prime to find #
# 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;
PR read "primes.incl.a68" PR                   # include prime utilities #
[]BOOL prime = PRIMESIEVE max prime;     # sieve the primes to max prime #
INT max show = 2 500;
# find the Erdős primes, showing the ones up to max show                 #
INT e count := 0;
IF is erdos prime( 2 ) THEN
print( ( " 2" ) );
e count +:= 1
FI;
INT last erdos := 0;
FOR p FROM 3 BY 2 TO max show DO
IF is erdos prime( p ) THEN
print( ( " ", whole( p, 0 ) ) );
last erdos := p;
e count   +:= 1
FI
OD;
print( ( newline, "Found ", whole( e count, 0 )
, " Erdos primes up to ", whole( max show, 0 ), newline ) );
# find the max erdos'th Erdős prime #
FOR p FROM max show WHILE e count < max erdos DO
IF is erdos prime( p ) THEN
last erdos := p;
e count   +:= 1
FI
OD;
print( ( whole( last erdos, 0 ), " is Erdos prime ", whole( e count, 0 ), newline ) )
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 up to 2500
999721 is Erdos prime 7875
```

## APL

```erdos_primes←{
prime ← {(⍵≥2) ∧ 0∧.≠(1↓⍳⌊⍵*÷2)|⍵}
erdos ← {(prime ⍵) ∧ ∧/~prime¨ ⍵-!⍳⌊(!⍣¯1)⍵}
e2500 ← (erdos¨e)/e←⍳2500
⎕←e2500
⎕←'There are ',(⍕⍴e2500),' Erdős numbers ≤ 2500'
}
```
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
There are 25 Erdős numbers ≤ 2500```

## Arturo

```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```

## AWK

Translation of: FreeBASIC
```# 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
```

## BASIC

### FreeBASIC

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

```

### Palo Alto Tiny BASIC

Translation of: Tiny BASIC – Removed overlapping loops.

Without the stretch goal because numbers are limited to signed 16-bit integers.

```10 REM ERDOS-PRIMES
20 LET P=2,C=1
30 PRINT C," ",P
40 FOR P=3 TO 2500 STEP 2
50 LET Z=P;GOSUB 1000
60 IF A=0 GOTO 160
70 REM F = K!
80 LET K=1,F=1,Z=P-F
90 IF Z<0 GOTO 150
100 GOSUB 1000
110 IF A=1 GOTO 150
120 LET K=K+1,F=F*K,Z=P-F
130 IF Z<0 GOTO 150
140 GOTO 100
150 IF Z<0 LET C=C+1;PRINT C," ",P
160 NEXT P
170 STOP
980 REM PRIMALITY OF Z BY TRIAL DIVISION
990 REM RESULT IS IN A
1000 LET A=0
1010 IF Z=2 LET A=1;RETURN
1020 IF Z<3 RETURN
1030 LET Y=2
1040 IF (Z/Y)*Y=Z RETURN
1050 IF Y*Y>=Z LET A=1;RETURN
1060 LET Y=Y+1
1070 GOTO 1040
1080 RETURN
```
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
```

### Tiny BASIC

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

Works with: TinyBasic

Tiny BASICs other than Tom Pittman's TinyBasic use `,` instead of `;` for string concatenation in `PRINT`.

```10 REM Erdős-primes
20 LET P = 1
30 LET C = 0
40 IF P > 2 THEN LET P = P + 2
50 IF P < 3 THEN LET P = P + 1
60 LET Z = P
70 GOSUB 1000
80 IF A = 0 THEN GOTO 40
90 LET K = 0
100 LET F = 1
110 LET K = K + 1
120 REM F = K!
130 LET F = F * K
140 LET Z = P - F
150 IF Z < 0 THEN GOTO 190
160 GOSUB 1000
170 IF A = 1 THEN GOTO 40
180 GOTO 110
190 LET C = C + 1
200 IF P < 2500 THEN PRINT C; "  "; P
210 IF P > 2500 THEN END
220 GOTO 40
990 REM primality of Z by trial division, result is in A
1000 LET Y = 1
1010 LET A = 0
1020 IF Z = 2 THEN LET A = 1
1030 IF Z < 3 THEN RETURN
1040 LET Y = Y + 1
1050 IF (Z / Y) * Y = Z THEN RETURN
1060 IF Y * Y < Z THEN GOTO 1040
1070 LET A = 1
1080 RETURN
```
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
```

## C

Translation of: Wren
```#include <stdio.h>
#include <stdlib.h>
#include <stdbool.h>
#include <locale.h>

#define LIMIT 1000000
#define LOWER_LIMIT 2500

bool *sieve(int limit) {
int i, p;
limit++;
// True denotes composite, false denotes prime.
bool *c = calloc(limit, sizeof(bool)); // 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.
while (true) {
int p2 = p * p;
if (p2 >= limit) break;
for (i = p2; i < limit; i += 2 * p) c[i] = true;
while (true) {
p += 2;
if (!c[p]) break;
}
}
return c;
}

int main() {
int i, j, fact, ec = 0, ec2 = 0, lastErdos = 0;
bool found;
bool *c = sieve(LIMIT-1);
int erdos[30];
for (i = 2; i < LIMIT;) {
if (!c[i]) {
j = 1;
fact = 1;
found = true;
while (fact < i) {
if (!c[i-fact]) {
found = false;
break;
}
++j;
fact *= j;
}
if (found) {
if (i < LOWER_LIMIT) erdos[ec2++] = i;
lastErdos = i;
++ec;
}
}
i = (i > 2) ? i + 2 : i + 1;
}
setlocale(LC_NUMERIC, "");
printf("The %'d Erdős primes under %'d are:\n", ec2, LOWER_LIMIT);
for (i = 0; i < ec2; ++i) {
printf("%6d ", erdos[i]);
if (!((i+1)%10)) printf("\n");
}
printf("\n\nThe %'dth Erdős prime is %'d.\n", ec, lastErdos);
free(c);
return 0;
}
```
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.
```

## C#

```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++

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.
```

## Delphi

Works with: Delphi version 6.0

Executes in 225 ms. It could be faster with a Factorial lookup table.

```function IsPrime(N: integer): boolean;
{Optimised prime test - about 40% faster than the naive approach}
var I,Stop: integer;
begin
if (N = 2) or (N=3) then Result:=true
else if (n <= 1) or ((n mod 2) = 0) or ((n mod 3) = 0) then Result:= false
else
begin
I:=5;
Stop:=Trunc(sqrt(N));
Result:=False;
while I<=Stop do
begin
if ((N mod I) = 0) or ((N mod (i + 2)) = 0) then exit;
Inc(I,6);
end;
Result:=True;
end;
end;

function Factorial(N: Word): int64;
var  I: integer;
begin
Result:= 1;
for I := 2 to N do Result:=Result * I;
end;

function IsErdosPrime(P: integer): boolean;
{Test if specified Primes is Erdos}
{i.e. all p-k! for 1<=k!<p are composite.}
var K: integer;
var F: int64;
begin
K:=1;
Result:=False;
while True do
begin
F:=Factorial(K);
if F>=P then break;
if IsPrime(P-F) then exit;
Inc(K);
end;
Result:=True;
end;

procedure FindErdosPrimes(Memo: TMemo);
{Collect and display Erdos primes}
var P,I,Cnt: integer;
var Erdos: array of integer;
var S: string;
begin
{Collect all Erdos Primes<1,000,000}
for P:=2 to 1000000 do
if IsPrime(P) then
if IsErdosPrime(P) then
begin
SetLength(Erdos,Length(Erdos)+1);
Erdos[High(Erdos)]:=P;
end;
{Display the data in several different ways}
Memo.Lines.Add('25 Erdos primes less than 2500');
S:='';
for I:=0 to 24 do
begin
S:=S+Format('%8d',[Erdos[I]]);
if (((I+1) mod 5)=0) or (I=24) then
begin
S:='';
end;
end;
Memo.Lines.Add('Number of Erdos Primes < 1-million: '+IntToStr(Length(Erdos)));
Memo.Lines.Add('Last Erdos Prime < 1-million: '+IntToStr(Erdos[High(Erdos)]));
end;
```
Output:
```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
Summary:
Number of Erdos Primes < 1-million: 7875
Last Erdos Prime < 1-million: 999721

```

## EasyLang

Translation of: Action!
```fastfunc isprim num .
if num < 2
return 0
.
i = 2
while i <= sqrt num
if num mod i = 0
return 0
.
i += 1
.
return 1
.
func iserdosprim p .
if isprim p = 0
return 0
.
k = 1
f = 1
while f < p
if isprim (p - f) = 1
return 0
.
k += 1
f *= k
.
return 1
.
for p = 2 to 2499
if iserdosprim p = 1
write p & " "
.
.```
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
```

## F#

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
```

## Factor

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.
```

## Forth

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
```

## Go

Translation of: Wren
Library: Go-rcu
```package main

import (
"fmt"
"rcu"
)

func main() {
limit := int(1e6)
lowerLimit := 2500
c := rcu.PrimeSieve(limit-1, true)
var erdos []int
lastErdos := 0
ec := 0
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 {
if i < lowerLimit {
erdos = append(erdos, i)
}
lastErdos = i
ec++
}
}
if i > 2 {
i += 2
} else {
i += 1
}
}
fmt.Printf("The %d Erdős primes under %s are\n", len(erdos), rcu.Commatize(lowerLimit))
rcu.PrintTable(erdos, 10, 6, false)
fmt.Printf("\nThe %s Erdős prime is %s.\n", rcu.Commatize(ec), rcu.Commatize(lastErdos))
}
```
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.
```

## J

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 }}
```

```   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
```

## Java

```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.
```

## jq

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 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.
```

## Julia

```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
```

## Lua

```function isPrime (x)
if x < 2 then return false end
if x < 4 then return true end
if x % 2 == 0 then return false end
for d = 3, math.sqrt(x), 2 do
if x % d == 0 then return false end
end
return true
end

function primes ()
local x, maxInt = 3, 2^53
local function yieldPrimes ()
coroutine.yield(2)
repeat
if isPrime(x) then coroutine.yield(x) end
x = x + 2
until x == maxInt
end
return coroutine.wrap(function() yieldPrimes() end)
end

function factorial (n)
local f = 1
for i = 2, n do
f = f * i
end
return f
end

function isErdos (p)
local k, factK = 1
repeat
factK = factorial(k)
if isPrime(p - factK) then return false end
k = k + 1
until factK >= p
return true
end

local nextPrime, count, prime = primes(), 0
while count < 7875 do
prime = nextPrime()
if isErdos(prime) then
if prime < 2500 then io.write(prime .. " ") end
count = count + 1
end
end
print("\n\nThe 7875th Erdos prime is " .. prime)
```
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

The 7875th Erdos prime is 999721```

## Mathematica/Wolfram Language

```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}```

## Miranda

```main :: [sys_message]
main = [Stdout (lay (map show erdos2500)),
Stdout ("There are " ++ show (#erdos2500) ++ " Erdos numbers <2500\n")]
where erdos2500 = filter erdos [1..2499]

erdos :: num->bool
erdos p = prime p & ~or [prime (p-k) | k <- takewhile (<p) (scan (*) 1 [2..])]

prime :: num->bool
prime n = n=2 \/ n=3, if n<=4
prime n = False, if n mod 2=0
prime n = and [n mod d ~= 0 | d <- [2..entier (sqrt n)]]```
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
There are 25 Erdos numbers <2500```

## Nim

```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.```

## Perl

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```

## Phix

```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)
```

## PL/0

Without stretch goal.

```var p, c, z, k, isprime, factk, iskchecked;

procedure checkprimality;
var i, isichecked;
begin
isprime := 0;
if z = 2 then isprime := 1;
if z >= 3 then
begin
i := 2; isichecked := 0;
while isichecked = 0 do
begin
if (z / i) * i = z then isichecked := 1;
if isichecked = 0 then
if i * i >= z then
begin
isprime := 1; isichecked := 1
end;
i := i + 1
end
end
end;

begin
p := 2; c := 1;
! p;
p := 3;
while p <= 2500 do
begin
z := p; call checkprimality;
if isprime = 1 then
begin
k := 1; factk := 1; z := p - factk;
iskchecked := 1;
if z >= 0 then iskchecked := 0;
while iskchecked = 0 do
begin
call checkprimality;
if isprime = 1 then iskchecked := 1;
if isprime <> 1 then
begin
k := k + 1; factk := factk * k; z := p - factk;
iskchecked := 1;
if z >= 0 then iskchecked := 0
end
end;
if z < 0 then
begin
c := c + 1;
! p
end
end;
p := p + 2
end
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
```

## PL/M

Works with: 8080 PL/M Compiler

... under CP/M (or an emulator)

Basic task only as PL/M integers are 8/16 bit unsigned.

```100H: /* FIND ERDOS PRIMES: PRIMES P WHERE P-K! IS COMPOSITE FOR ALL 1<=K!<P */

/* CP/M SYSTEM CALL AND I/O ROUTINES                                      */
BDOS:      PROCEDURE( FN, ARG ); DECLARE FN BYTE, ARG ADDRESS; GOTO 5; END;
PR\$CHAR:   PROCEDURE( C ); DECLARE C BYTE;    CALL BDOS( 2, C );  END;
PR\$STRING: PROCEDURE( S ); DECLARE S ADDRESS; CALL BDOS( 9, S );  END;
PR\$NL:     PROCEDURE;   CALL PR\$CHAR( 0DH ); CALL PR\$CHAR( 0AH ); END;
PR\$NUMBER: PROCEDURE( N ); /* PRINTS A NUMBER IN THE MINIMUN FIELD WIDTH  */
DECLARE V ADDRESS, N\$STR ( 6 )BYTE, W BYTE;
V = N;
W = LAST( N\$STR );
N\$STR( W ) = '\$';
N\$STR( W := W - 1 ) = '0' + ( V MOD 10 );
DO WHILE( ( V := V / 10 ) > 0 );
N\$STR( W := W - 1 ) = '0' + ( V MOD 10 );
END;
CALL PR\$STRING( .N\$STR( W ) );
END PR\$NUMBER;

DECLARE MAX\$NUMBER LITERALLY '2500'
, FALSE      LITERALLY '0'
, TRUE       LITERALLY '0FFH'
;

DECLARE PRIME ( MAX\$NUMBER )BYTE;   /* SIEVE THE PRIMES TO MAX\$NUMBER - 1 */
PRIME( 0 ), PRIME( 1 ) = FALSE;
PRIME( 2 ) = TRUE;
DO I = 3 TO LAST( PRIME ) BY 2; PRIME( I ) = TRUE;  END;
DO I = 4 TO LAST( PRIME ) BY 2; PRIME( I ) = FALSE; END;
DO I = 3 TO LAST( PRIME ) BY 2;
IF PRIME( I ) THEN DO;
DO S = I + I TO LAST( PRIME ) BY I;
PRIME( S ) = FALSE;
END;
END;
END;

/* TABLE OF FACTORIALS                                                   */
DECLARE FACTORIAL ( 8 )ADDRESS INITIAL( 1, 1, 2, 6, 24, 120, 720, 5040 );

/* RETURNS TRUE IF P IS AN ERDOS PRIME, FALSE OTHERWISE                  */
IS\$ERDOS\$PRIME: PROCEDURE( P )BYTE;
DECLARE RESULT BYTE;
RESULT = PRIME( P );
IF RESULT THEN DO;
DECLARE K BYTE;
K = 1;
DO WHILE FACTORIAL( K ) < P AND RESULT;
RESULT = NOT PRIME( P - FACTORIAL( K ) );
K = K + 1;
END;
END;
RETURN RESULT;
END IS\$ERDOS\$PRIME ;

/* FIND THE ERDOS PRIMES                                                  */
DECLARE ( P, COUNT ) ADDRESS;
COUNT = 0;
IF IS\$ERDOS\$PRIME( 2 ) THEN DO;
COUNT = COUNT + 1;
CALL PR\$STRING( .' 2\$' );
END;
P = 1;
DO WHILE COUNT < 25;
P = P + 2;
IF IS\$ERDOS\$PRIME( P ) THEN DO;
COUNT = COUNT + 1;
CALL PR\$CHAR( ' ' );
IF P < 1000 THEN CALL PR\$CHAR( ' ' );
CALL PR\$NUMBER( P );
IF COUNT MOD 5 = 0 THEN CALL PR\$NL;
END;
END;

EOF```
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
```

## Raku

```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```

Alternately, using fewer hard coded values:

```use Lingua::EN::Numbers;
use List::Divvy;

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

put "Erdős primes < 2500:\n" ~ @Erdős.&before(2500)».&comma.batch(8)».fmt("%5s").join: "\n";
put "\nThe largest Erdős prime less than {comma 1e6.Int} is {comma .[*-1]} in {.&ordinal-digit} position."
given @Erdős.&before(1e6);
```
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 largest Erdős prime less than 1,000,000 is 999,721 in 7875th position.```

## REXX

```/*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.
```

## Ring

```load "stdlibcore.ring"
see "working..." + nl
row = 0
limit = 2500

for p = 1 to limit
flag = 1
if isprime(p)
for k = 1 to p
if factorial(k) < p
temp = p - factorial(k)
if not isprime(temp)
flag = 1
else
flag = 0
exit
ok
else
exit
ok
next
else
flag = 0
ok
if flag = 1
row++
see "" + p + " "
if row % 5 = 0
see nl
ok
ok
next

see nl + "Found " + row + " Erdos primes less than 2500" + nl
see "done..." + nl```
Output:
```working...
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 less than 2500
done...
```

## RPL

Works with: HP version 49
```« 0 → p k
« 1 SF
WHILE p 'k' INCR FACT - DUP 0 > 1 FS? AND REPEAT
IF ISPRIME? THEN 1 CF END
END
DROP 1 FS?
» » 'ERDOS?' STO      @  ( n → erdös(n) )

« { } 2
WHILE DUP 2500 < REPEAT
IF DUP ERDOS? THEN SWAP OVER + SWAP END
NEXTPRIME
END
DROP DUP SIZE "count" →TAG
» 'TASK' STO        @   ( → results )
```
Output:
```2: { 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 }
1: count: 25.
```

## Rust

```// [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.
```

## SETL

```program erdos_primes;
loop for e in [1..2499] | erdos e do
if (n +:= 1) mod 10=0 then print; end if;
end loop;

print;
print("There are " + str n + " Erdos numbers < 2500");

e := 2499;
loop while n < 7875 do
loop until erdos e do
e +:= 2;
end loop;
n +:= 1;
end loop;

print("The " + str n + "th Erdos number is " + str e);

op erdos(p);
return prime p and not exists k in faclist p | prime (p-k);
end erdos;

op faclist(n);
f := 1;
return [[i+:=1, f*:=i](2) : until n<f](..i-1);
end op;

op prime(n);
if n<=4 then
return n in [2,3];
end if;
return odd n and not exists d in [3, 5..floor (sqrt n)] | n mod d=0;
end op;
end program;```
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
There are 25 Erdos numbers < 2500
The 7875th Erdos number is 999721```

## Sidef

```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
```

## Wren

Library: Wren-math
Library: Wren-fmt
```import "./math" for Int
import "./fmt" for Fmt

var limit = 1e6
var lowerLimit = 2500
var c = Int.primeSieve(limit - 1, false)
var erdos = []
var lastErdos = 0
var ec = 0
var i = 2
while (i < limit) {
if (!c[i]) {
var j = 1
var fact = 1
var found = true
while (fact < i) {
if (!c[i - fact]) {
found = false
break
}
j = j + 1
fact = fact * j
}
if (found) {
lastErdos = i
ec = ec + 1
}
}
i = (i > 2) ? i + 2 : i + 1
}

Fmt.print("The \$,d Erdős primes under \$,d are:", erdos.count, lowerLimit)
Fmt.tprint("\$6d", erdos, 10)
Fmt.print("\nThe \$,r Erdős prime is \$,d.", ec, lastErdos)
```
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.
```

## XPL0

```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
```