Strange unique prime triplets
Primes n, m, and p are strange unique primes if n, m, and p are unique and their sum n + m + p is also prime.
- Task
-
- Find all triplets of strange unique primes in which n, m, and p are all less than 30.
- (stretch goal) Show the count (only) of all the triplets of strange unique primes in which n, m, and p are all less than 1,000.
ALGOL W
Based on
<lang algolw>begin % find some strange unique primes - triplets of primes n, m, p %
% where n + m + p is also prime and n =/= m =/= p % % sets p( 1 :: n ) to a sieve of primes up to n % procedure Eratosthenes ( logical array p( * ) ; integer value n ) ; begin p( 1 ) := false; p( 2 ) := true; for i := 3 step 2 until n do p( i ) := true; for i := 4 step 2 until n do p( i ) := false; for i := 3 step 2 until truncate( sqrt( n ) ) do begin integer ii; ii := i + i; if p( i ) then for pr := i * i step ii until n do p( pr ) := false end for_i ; end Eratosthenes ; % we need to find the strange unique prime triplets below 1000 % integer MAX_PRIME; MAX_PRIME := 1000; begin % the sum of the triplets could be (roughly) 3 x the largest prime % logical array p ( 1 :: MAX_PRIME * 3 ); integer sCount, c30; % construct a sieve of primes up to MAX_PRIME * 3 % Eratosthenes( p, MAX_PRIME * 3 ); % count the nice prime triplets whose members are less than 1000 % % and prime the first 30 % sCount := c30 := 0; % 2 cannot be one of the primes as the sum would be even otherwise % for n := 3 step 2 until MAX_PRIME - 5 do begin if p( n ) then begin for m := n + 2 step 2 until MAX_PRIME - 3 do begin if p( m ) then begin for l := m + 2 STEP 2 until MAX_PRIME do begin if p( l ) then begin integer s; s := n + m + l; if p( s ) then begin sCount := sCount + 1; if l <= 30 and m <= 30 and n <= 30 then begin c30 := c30 + 1; write( i_w := 3, s_w := 0, c30, ": ", n, " + ", m, " + ", l, " = ", s ) end if_l_m_n_le_30 end if_p_s end if_p_l end for_l end if_p_m end for_m end if_p_n end for_n ; write( i_w := 3, s_w := 0, "Found ", c30, " strange unique prime triplets up to 30" ); write( i_w := 3, s_w := 0, "Found ", sCount, " strange unique prime triplets up to 1000" ); end
end.</lang>
- Output:
1: 3 + 5 + 11 = 19 2: 3 + 5 + 23 = 31 3: 3 + 5 + 29 = 37 4: 3 + 7 + 13 = 23 5: 3 + 7 + 19 = 29 6: 3 + 11 + 17 = 31 7: 3 + 11 + 23 = 37 8: 3 + 11 + 29 = 43 9: 3 + 17 + 23 = 43 10: 5 + 7 + 11 = 23 11: 5 + 7 + 17 = 29 12: 5 + 7 + 19 = 31 13: 5 + 7 + 29 = 41 14: 5 + 11 + 13 = 29 15: 5 + 13 + 19 = 37 16: 5 + 13 + 23 = 41 17: 5 + 13 + 29 = 47 18: 5 + 17 + 19 = 41 19: 5 + 19 + 23 = 47 20: 5 + 19 + 29 = 53 21: 7 + 11 + 13 = 31 22: 7 + 11 + 19 = 37 23: 7 + 11 + 23 = 41 24: 7 + 11 + 29 = 47 25: 7 + 13 + 17 = 37 26: 7 + 13 + 23 = 43 27: 7 + 17 + 19 = 43 28: 7 + 17 + 23 = 47 29: 7 + 17 + 29 = 53 30: 7 + 23 + 29 = 59 31: 11 + 13 + 17 = 41 32: 11 + 13 + 19 = 43 33: 11 + 13 + 23 = 47 34: 11 + 13 + 29 = 53 35: 11 + 17 + 19 = 47 36: 11 + 19 + 23 = 53 37: 11 + 19 + 29 = 59 38: 13 + 17 + 23 = 53 39: 13 + 17 + 29 = 59 40: 13 + 19 + 29 = 61 41: 17 + 19 + 23 = 59 42: 19 + 23 + 29 = 71 Found 42 strange unique prime triplets up to 30 Found 241580 strange unique prime triplets up to 1000
Delphi
<lang Delphi> program Strange_primes;
{$APPTYPE CONSOLE}
uses
System.SysUtils;
function IsPrime(n: Integer): Boolean; begin
if n < 2 then exit(false);
if n mod 2 = 0 then exit(n = 2);
if n mod 3 = 0 then exit(n = 3);
var d := 5; while d * d <= n do begin if n mod d = 0 then exit(false);
inc(d, 2);
if n mod d = 0 then exit(false);
inc(d, 4); end; Result := true;
end;
function Commatize(value: Integer): string; begin
Result := FloatToStrF(value, ffNumber, 10, 0);
end;
function StrangePrimes(n: Integer; countOnly: Boolean): Integer; begin
var c := 0; var f := '%2d: %2d + %2d + %2d = %2d'#10; var s: Integer := 0;
var i := 3; while i <= n - 4 do begin if IsPrime(i) then begin var j := i + 2; while j <= n - 2 do begin if IsPrime(j) then begin var k := j + 2; while k <= n do begin if IsPrime(k) then begin s := i + j + k; if IsPrime(s) then begin inc(c); if not countOnly then write(format(f, [c, i, j, k, s])); end; end; inc(k, 2); end; end; inc(j, 2); end; end; inc(i, 2); end; Result := c;
end;
begin
Writeln('Unique prime triples under 30 which sum to a prime:'); strangePrimes(29, false); var cs := commatize(strangePrimes(999, true)); writeln('There are ', cs, ' unique prime triples under 1,000 which sum to a prime.'); readln;
end.</lang>
Factor
<lang factor>USING: formatting io kernel math math.combinatorics math.primes sequences tools.memory.private ;
- .triplet ( seq -- ) "%2d+%2d+%2d = %d\n" vprintf ;
- strange ( n -- )
primes-upto 3 [ dup sum dup prime? [ suffix .triplet ] [ 2drop ] if ] each-combination ;
- count-strange ( n -- count )
0 swap primes-upto 3 [ sum prime? [ 1 + ] when ] each-combination ;
30 strange 1,000 count-strange commas nl "Found %s strange prime triplets with n, m, p < 1,000.\n" printf</lang>
- Output:
3+ 5+11 = 19 3+ 5+23 = 31 3+ 5+29 = 37 3+ 7+13 = 23 3+ 7+19 = 29 3+11+17 = 31 3+11+23 = 37 3+11+29 = 43 3+17+23 = 43 5+ 7+11 = 23 5+ 7+17 = 29 5+ 7+19 = 31 5+ 7+29 = 41 5+11+13 = 29 5+13+19 = 37 5+13+23 = 41 5+13+29 = 47 5+17+19 = 41 5+19+23 = 47 5+19+29 = 53 7+11+13 = 31 7+11+19 = 37 7+11+23 = 41 7+11+29 = 47 7+13+17 = 37 7+13+23 = 43 7+17+19 = 43 7+17+23 = 47 7+17+29 = 53 7+23+29 = 59 11+13+17 = 41 11+13+19 = 43 11+13+23 = 47 11+13+29 = 53 11+17+19 = 47 11+19+23 = 53 11+19+29 = 59 13+17+23 = 53 13+17+29 = 59 13+19+29 = 61 17+19+23 = 59 19+23+29 = 71 Found 241,580 strange prime triplets with n, m, p < 1,000.
Go
Basic
<lang go>package main
import "fmt"
func isPrime(n int) bool {
switch { case n < 2: return false case n%2 == 0: return n == 2 case n%3 == 0: return n == 3 default: d := 5 for d*d <= n { if n%d == 0 { return false } d += 2 if n%d == 0 { return false } d += 4 } return true }
}
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 strangePrimes(n int, countOnly bool) int {
c := 0 f := "%2d: %2d + %2d + %2d = %2d\n" var s int
for i := 3; i <= n-4; i += 2 { if isPrime(i) { for j := i + 2; j <= n-2; j += 2 { if isPrime(j) { for k := j + 2; k <= n; k += 2 { if isPrime(k) { s = i + j + k if isPrime(s) { c++ if !countOnly { fmt.Printf(f, c, i, j, k, s) } } } } } } } } return c
}
func main() {
fmt.Println("Unique prime triples under 30 which sum to a prime:") strangePrimes(29, false) cs := commatize(strangePrimes(999, true)) fmt.Printf("\nThere are %s unique prime triples under 1,000 which sum to a prime.\n", cs)
}</lang>
- Output:
Unique prime triples under 30 which sum to a prime: 1: 3 + 5 + 11 = 19 2: 3 + 5 + 23 = 31 3: 3 + 5 + 29 = 37 4: 3 + 7 + 13 = 23 5: 3 + 7 + 19 = 29 6: 3 + 11 + 17 = 31 7: 3 + 11 + 23 = 37 8: 3 + 11 + 29 = 43 9: 3 + 17 + 23 = 43 10: 5 + 7 + 11 = 23 11: 5 + 7 + 17 = 29 12: 5 + 7 + 19 = 31 13: 5 + 7 + 29 = 41 14: 5 + 11 + 13 = 29 15: 5 + 13 + 19 = 37 16: 5 + 13 + 23 = 41 17: 5 + 13 + 29 = 47 18: 5 + 17 + 19 = 41 19: 5 + 19 + 23 = 47 20: 5 + 19 + 29 = 53 21: 7 + 11 + 13 = 31 22: 7 + 11 + 19 = 37 23: 7 + 11 + 23 = 41 24: 7 + 11 + 29 = 47 25: 7 + 13 + 17 = 37 26: 7 + 13 + 23 = 43 27: 7 + 17 + 19 = 43 28: 7 + 17 + 23 = 47 29: 7 + 17 + 29 = 53 30: 7 + 23 + 29 = 59 31: 11 + 13 + 17 = 41 32: 11 + 13 + 19 = 43 33: 11 + 13 + 23 = 47 34: 11 + 13 + 29 = 53 35: 11 + 17 + 19 = 47 36: 11 + 19 + 23 = 53 37: 11 + 19 + 29 = 59 38: 13 + 17 + 23 = 53 39: 13 + 17 + 29 = 59 40: 13 + 19 + 29 = 61 41: 17 + 19 + 23 = 59 42: 19 + 23 + 29 = 71 There are 241,580 unique prime triples under 1,000 which sum to a prime.
Faster
<lang go>package main
import "fmt"
var sieved []bool var p = []int{2}
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 // no need to bother with even numbers over 2 for this task 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 strangePrimes(n int, countOnly bool) int {
c := 0 f := "%2d: %2d + %2d + %2d = %2d\n" var r, s int m := 0 for ; m < len(p) && p[m] <= n; m++ { } for i := 1; i < m-2; i++ { for j := i + 1; j < m-1; j++ { r = p[i] + p[j] for k := j + 1; k < m; k++ { s = r + p[k] if !sieved[s] { c++ if !countOnly { fmt.Printf(f, c, p[i], p[j], p[k], s) } } } } } return c
}
func main() {
const max = 1000 sieved = sieve(3*max) for i := 3; i <= max; i += 2 { if !sieved[i] { p = append(p, i) } } fmt.Println("Unique prime triples under 30 which sum to a prime:") strangePrimes(29, false) cs := commatize(strangePrimes(999, true)) fmt.Printf("\nThere are %s unique prime triples under 1,000 which sum to a prime.\n", cs)
}</lang>
- Output:
Same as 'basic' version.
Julia
<lang julia>using Primes
function prime_sum_prime_triplets_to(N, verbose=false)
a = primes(3, N) prime_sieve_set = primesmask(1, N * 3) len, triplets, n = length(a), Dict{Tuple{Int64,Int64,Int64}, Int}(), 0 for i in eachindex(a), j in i+1:len, k in j+1:len if prime_sieve_set[a[i] + a[j] + a[k]] verbose && (triplets[(a[i], a[j], a[k])] = 1) n += 1 end end if verbose len = (length(string(N)) + 2) * 3 println("\n", rpad("Triplet", len), "Sum\n", "-"^(len+3)) for k in sort(collect(keys(triplets)), lt = (x, y) -> collect(x) < collect(y)) println(rpad(k, len), sum(k)) end end println("\n\n$n unique triplets of 3 primes between 2 and $N sum to a prime.") return triplets
end
prime_sum_prime_triplets_to(30, true) prime_sum_prime_triplets_to(1000) @time prime_sum_prime_triplets_to(10000) @time prime_sum_prime_triplets_to(100000)
</lang>
- Output:
Triplet Sum --------------- (3, 5, 11) 19 (3, 5, 23) 31 (3, 5, 29) 37 (3, 7, 13) 23 (3, 7, 19) 29 (3, 11, 17) 31 (3, 11, 23) 37 (3, 11, 29) 43 (3, 17, 23) 43 (5, 7, 11) 23 (5, 7, 17) 29 (5, 7, 19) 31 (5, 7, 29) 41 (5, 11, 13) 29 (5, 13, 19) 37 (5, 13, 23) 41 (5, 13, 29) 47 (5, 17, 19) 41 (5, 19, 23) 47 (5, 19, 29) 53 (7, 11, 13) 31 (7, 11, 19) 37 (7, 11, 23) 41 (7, 11, 29) 47 (7, 13, 17) 37 (7, 13, 23) 43 (7, 17, 19) 43 (7, 17, 23) 47 (7, 17, 29) 53 (7, 23, 29) 59 (11, 13, 17)41 (11, 13, 19)43 (11, 13, 23)47 (11, 13, 29)53 (11, 17, 19)47 (11, 19, 23)53 (11, 19, 29)59 (13, 17, 23)53 (13, 17, 29)59 (13, 19, 29)61 (17, 19, 23)59 (19, 23, 29)71 42 unique triplets of 3 primes between 2 and 30 sum to a prime. 241580 unique triplets of 3 primes between 2 and 1000 sum to a prime. 74588542 unique triplets of 3 primes between 2 and 10000 sum to a prime. 0.509732 seconds (31 allocations: 25.938 KiB) 28694800655 unique triplets of 3 primes between 2 and 100000 sum to a prime. 224.940756 seconds (35 allocations: 218.156 KiB)
Phix
<lang Phix>requires("0.8.4") function create_sieve(integer limit)
sequence sieve = repeat(true,limit) sieve[1] = false for i=4 to limit by 2 do sieve[i] = false end for for p=3 to floor(sqrt(limit)) by 2 do integer p2 = p*p if sieve[p2] then for k=p2 to limit by p*2 do sieve[k] = false end for end if end for return sieve
end function
procedure strange_triplets(integer lim, bool bCountOnly=true)
atom t0 = time(), t1 = t0+1 sequence primes = get_primes_le(lim), sieve = create_sieve(lim*3), res = {} atom count = 0 -- -- It is not worth involving 2, ie primes[1], -- since (2 + any other two primes) is even, -- also we may as well leave space for {j,k}, -- {k} in the two outer loops. -- Using a sieve on the inner test is over -- ten times faster than is_prime(), whereas -- using a separate table of primes for the -- two outer loops is about twice as fast as -- scanning the sieve skipping falsies. Also -- interestingly, using nm = n+m is twice as -- fast as nmp = n+m+p. -- for i=2 to length(primes)-2 do integer n = primes[i] for j=i+1 to length(primes)-1 do integer m = primes[j], nm = n+m for k=j+1 to length(primes) do integer p = primes[k], nmp = nm+p if sieve[nmp] then count += 1 if not bCountOnly then res = append(res,sprintf("%2d: %2d+%2d+%2d = %d", {count, n, m, p, nmp})) end if end if if time()>t1 then progress("Working... (%,d)\r",{count}) t1 = time()+1 end if end for end for end for progress("") string r = iff(bCountOnly?sprintf(" (%s)",{elapsed(time()-t0)}) :sprintf(":\n%s",{join(shorten(res,"",3),"\n")})) printf(1,"%,d strange triplets < %,d found%s\n\n",{count,lim,r})
end procedure
strange_triplets(30,false) strange_triplets(1000) strange_triplets(10000)</lang>
- Output:
42 strange triplets < 30 found: 1: 3+ 5+11 = 19 2: 3+ 5+23 = 31 3: 3+ 5+29 = 37 ... 40: 13+19+29 = 61 41: 17+19+23 = 59 42: 19+23+29 = 71 241,580 strange triplets < 1,000 found (0.0s) 74,588,542 strange triplets < 10,000 found (11.4s)
Python
Using sympy.primerange.
<lang python>from sympy import primerange
def strange_triplets(mx: int = 30) -> None:
primes = list(primerange(0, mx)) primes3 = set(primerange(0, 3 * mx)) for i, n in enumerate(primes): for j, m in enumerate(primes[i + 1:], i + 1): for p in primes[j + 1:]: if n + m + p in primes3: yield n, m, p
for c, (n, m, p) in enumerate(strange_triplets(), 1):
print(f"{c:2}: {n:2}+{m:2}+{p:2} = {n + m + p}")
mx = 1_000 print(f"\nIf n, m, p < {mx:_} finds {sum(1 for _ in strange_triplets(mx)):_}")</lang>
- Output:
1: 3+ 5+11 = 19 2: 3+ 5+23 = 31 3: 3+ 5+29 = 37 4: 3+ 7+13 = 23 5: 3+ 7+19 = 29 6: 3+11+17 = 31 7: 3+11+23 = 37 8: 3+11+29 = 43 9: 3+17+23 = 43 10: 5+ 7+11 = 23 11: 5+ 7+17 = 29 12: 5+ 7+19 = 31 13: 5+ 7+29 = 41 14: 5+11+13 = 29 15: 5+13+19 = 37 16: 5+13+23 = 41 17: 5+13+29 = 47 18: 5+17+19 = 41 19: 5+19+23 = 47 20: 5+19+29 = 53 21: 7+11+13 = 31 22: 7+11+19 = 37 23: 7+11+23 = 41 24: 7+11+29 = 47 25: 7+13+17 = 37 26: 7+13+23 = 43 27: 7+17+19 = 43 28: 7+17+23 = 47 29: 7+17+29 = 53 30: 7+23+29 = 59 31: 11+13+17 = 41 32: 11+13+19 = 43 33: 11+13+23 = 47 34: 11+13+29 = 53 35: 11+17+19 = 47 36: 11+19+23 = 53 37: 11+19+29 = 59 38: 13+17+23 = 53 39: 13+17+29 = 59 40: 13+19+29 = 61 41: 17+19+23 = 59 42: 19+23+29 = 71 If n, m, p < 1_000 finds 241_580
REXX
<lang rexx>/*REXX program finds/lists triplet strange primes (<HI) where the triplets' sum is prime*/ parse arg hi . /*obtain optional argument from the CL.*/ if hi== | hi=="," then hi= 30 /*Not specified? Then use the default.*/ tell= hi>0; hi= abs(hi); hi= hi - 1 /*use absolute value of HI for limit. */ if tell>0 then say 'list of unique triplet strange primes whose sum is a prime.:' call genP /*build array of semaphores for primes.*/ finds= 0 /*# of triplet strange primes (so far).*/ say
do m=2+1 by 2 to hi; if \!.m then iterate /*just use the odd primes. */ do n=m+2 by 2 to hi; if \!.n then iterate /* " " " " " */ mn= m + n /*partial sum (deep loops).*/ do p=n+2 by 2 to hi; if \!.p then iterate /*just use the odd primes. */ sum= mn + p /*compute sum of 3 primes. */ if \!.sum then iterate /*Is the sum prime? No, then skip it.*/ finds= finds + 1 /*bump # of triplet "strange" primes.*/ if tell then say right(m, w+9) right(n, w) right(p, w) ' sum to:' right(sum, w+2) end /*p*/ end /*n*/ end /*m*/
say say 'Found ' commas(finds) " unique triplet strange primes < " commas(hi+1) ,
" which sum to a 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 ? /*──────────────────────────────────────────────────────────────────────────────────────*/ genP: !.= 0; w= length(hi) /*placeholders for primes; width of #'s*/
@.1=2; @.2=3; @.3=5; @.4=7; @.5=11 /*define some low primes. */ !.2=1; !.3=1; !.5=1; !.7=1; !.11=1 /* " " " " flags. */ #=5; s.#= @.# **2 /*number of primes so far; prime². */ /* [↓] generate more primes ≤ high.*/ do j=@.#+2 by 2 for hi*3%2 /*find odd primes from here on. */ parse var j -1 _; if _==5 then iterate /*J divisible by 5? (right dig)*/ if j// 3==0 then iterate /*" " " 3? */ if j// 7==0 then iterate /*" " " 7? */ /* [↑] the above five lines saves time*/ do k=5 while s.k<=j /* [↓] divide by the known odd primes.*/ if j // @.k == 0 then iterate j /*Is J ÷ X? Then not prime. ___ */ end /*k*/ /* [↑] only process numbers ≤ √ J */ #= #+1; @.#= j; s.#= j*j; !.j= 1 /*bump # of Ps; assign next P; P²; P# */ end /*j*/; return</lang>
- output when using the default input:
list of unique triplet strange primes that sum to a prime: prime generation took 0.02 seconds. 3 5 11 sum to: 19 3 5 23 sum to: 31 3 5 29 sum to: 37 3 7 13 sum to: 23 3 7 19 sum to: 29 3 11 17 sum to: 31 3 11 23 sum to: 37 3 11 29 sum to: 43 3 17 23 sum to: 43 5 7 11 sum to: 23 5 7 17 sum to: 29 5 7 19 sum to: 31 5 7 29 sum to: 41 5 11 13 sum to: 29 5 13 19 sum to: 37 5 13 23 sum to: 41 5 13 29 sum to: 47 5 17 19 sum to: 41 5 19 23 sum to: 47 5 19 29 sum to: 53 7 11 13 sum to: 31 7 11 19 sum to: 37 7 11 23 sum to: 41 7 11 29 sum to: 47 7 13 17 sum to: 37 7 13 23 sum to: 43 7 17 19 sum to: 43 7 17 23 sum to: 47 7 17 29 sum to: 53 7 23 29 sum to: 59 11 13 17 sum to: 41 11 13 19 sum to: 43 11 13 23 sum to: 47 11 13 29 sum to: 53 11 17 19 sum to: 47 11 19 23 sum to: 53 11 19 29 sum to: 59 13 17 23 sum to: 53 13 17 29 sum to: 59 13 19 29 sum to: 61 17 19 23 sum to: 59 19 23 29 sum to: 71 Found 42 unique triplet strange primes < 30 which sum to a prime.
- output when using the input of: -1000
Found 241,580 unique triplet strange primes < 1,000 which sum to a prime.
Ring
<lang ring> load "stdlib.ring"
num = 0 limit = 30
see "working..." + nl see "the strange primes are:" + nl
for n = 1 to limit
for m = n+1 to limit for p = m+1 to limit sum = n+m+p if isprime(sum) and isprime(n) and isprime(m) and isprime(p) num = num + 1 see "" + num + ": " + n + "+" + m + "+" + p + " = " + sum + nl ok next next
next
see "done..." + nl </lang>
- Output:
working... the strange primes are: 1: 3+5+11 = 19 2: 3+5+23 = 31 3: 3+5+29 = 37 4: 3+7+13 = 23 5: 3+7+19 = 29 6: 3+11+17 = 31 7: 3+11+23 = 37 8: 3+11+29 = 43 9: 3+17+23 = 43 10: 5+7+11 = 23 11: 5+7+17 = 29 12: 5+7+19 = 31 13: 5+7+29 = 41 14: 5+11+13 = 29 15: 5+13+19 = 37 16: 5+13+23 = 41 17: 5+13+29 = 47 18: 5+17+19 = 41 19: 5+19+23 = 47 20: 5+19+29 = 53 21: 7+11+13 = 31 22: 7+11+19 = 37 23: 7+11+23 = 41 24: 7+11+29 = 47 25: 7+13+17 = 37 26: 7+13+23 = 43 27: 7+17+19 = 43 28: 7+17+23 = 47 29: 7+17+29 = 53 30: 7+23+29 = 59 31: 11+13+17 = 41 32: 11+13+19 = 43 33: 11+13+23 = 47 34: 11+13+29 = 53 35: 11+17+19 = 47 36: 11+19+23 = 53 37: 11+19+29 = 59 38: 13+17+23 = 53 39: 13+17+29 = 59 40: 13+19+29 = 61 41: 17+19+23 = 59 42: 19+23+29 = 71 done...
Wren
Basic
<lang ecmascript>import "/math" for Int import "/trait" for Stepped import "/fmt" for Fmt
var strangePrimes = Fn.new { |n, countOnly|
var c = 0 var s for (i in Stepped.new(3..n-4, 2)) { if (Int.isPrime(i)) { for (j in Stepped.new(i+2..n-2, 2)) { if (Int.isPrime(j)) { for (k in Stepped.new(j+2..n, 2)) { if (Int.isPrime(k) && Int.isPrime(s = i + j + k)) { c = c + 1 if (!countOnly) Fmt.print("$2d: $2d + $2d + $2d = $2d", c, i, j, k, s) } } } } } } return c
}
System.print("Unique prime triples under 30 which sum to a prime:") strangePrimes.call(29, false) var c = strangePrimes.call(999, true) Fmt.print("\nThere are $,d unique prime triples under 1,000 which sum to a prime.", c)</lang>
- Output:
Unique prime triples under 30 which sum to a prime: 1: 3 + 5 + 11 = 19 2: 3 + 5 + 23 = 31 3: 3 + 5 + 29 = 37 4: 3 + 7 + 13 = 23 5: 3 + 7 + 19 = 29 6: 3 + 11 + 17 = 31 7: 3 + 11 + 23 = 37 8: 3 + 11 + 29 = 43 9: 3 + 17 + 23 = 43 10: 5 + 7 + 11 = 23 11: 5 + 7 + 17 = 29 12: 5 + 7 + 19 = 31 13: 5 + 7 + 29 = 41 14: 5 + 11 + 13 = 29 15: 5 + 13 + 19 = 37 16: 5 + 13 + 23 = 41 17: 5 + 13 + 29 = 47 18: 5 + 17 + 19 = 41 19: 5 + 19 + 23 = 47 20: 5 + 19 + 29 = 53 21: 7 + 11 + 13 = 31 22: 7 + 11 + 19 = 37 23: 7 + 11 + 23 = 41 24: 7 + 11 + 29 = 47 25: 7 + 13 + 17 = 37 26: 7 + 13 + 23 = 43 27: 7 + 17 + 19 = 43 28: 7 + 17 + 23 = 47 29: 7 + 17 + 29 = 53 30: 7 + 23 + 29 = 59 31: 11 + 13 + 17 = 41 32: 11 + 13 + 19 = 43 33: 11 + 13 + 23 = 47 34: 11 + 13 + 29 = 53 35: 11 + 17 + 19 = 47 36: 11 + 19 + 23 = 53 37: 11 + 19 + 29 = 59 38: 13 + 17 + 23 = 53 39: 13 + 17 + 29 = 59 40: 13 + 19 + 29 = 61 41: 17 + 19 + 23 = 59 42: 19 + 23 + 29 = 71 There are 241,580 unique prime triples under 1,000 which sum to a prime.
Faster
The following version uses a prime sieve and is about 17 times faster than the 'basic' version. <lang ecmascript>import "/math" for Int import "/fmt" for Fmt
var max = 1000 var sieved = Int.primeSieve(3*max, false) // includes composites var p = Int.primeSieve(max, true) // primes only
var strangePrimes = Fn.new { |n, countOnly|
var c = 0 var m = 0 while (m < p.count && p[m] <= n) m = m + 1 var r var s for (i in 1...m-2) { for (j in i+1...m-1) { r = p[i] + p[j] for (k in j+1...m) { if (!sieved[s = r + p[k]]) { c = c + 1 if (!countOnly) Fmt.print("$2d: $2d + $2d + $2d = $2d", c, p[i], p[j], p[k], s) } } } } return c
}
System.print("Unique prime triples under 30 which sum to a prime:") strangePrimes.call(29, false) var c = strangePrimes.call(999, true) Fmt.print("\nThere are $,d unique prime triples under 1,000 which sum to a prime.", c)</lang>
- Output:
Same as 'basic' version.