Strange unique prime triplets
Integers n, m, and p are strange unique primes if n, m, and p are unique primes, and the sum of 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.
Factor
<lang factor>USING: formatting io kernel math math.combinatorics math.primes sequences tools.memory.private ;
- strange ( n -- seq )
primes-upto 3 [ sum prime? ] filter-combinations ;
30 strange [
1 + over sum [ prefix ] dip suffix "%2d: %2d+%2d+%2d = %d\n" vprintf
] each-index nl
1,000 strange length commas "Found %s strange prime triplets with n, m, p < 1,000.\n" printf</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 241,580 strange prime triplets with n, m, p < 1,000.
Go
<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.
Phix
<lang Phix>requires("0.8.4") constant primes = get_primes_le(30) --constant primes = get_primes(-10) -- earlier versions
procedure strange_triplets()
-- -- 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 -- sequence res = {} 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] for k=j+1 to length(primes) do integer p = primes[k], nmp = n+m+p if is_prime(nmp) then res = append(res,sprintf("%2d: %2d+%2d+%2d = %d", {length(res)+1, n, m, p, nmp})) end if end for end for end for printf(1,"%d found:\n%s\n",{length(res),join(shorten(res,"",3),"\n")})
end procedure
strange_triplets()</lang>
- Output:
42 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
With get_primes_le(1000), or get_primes(-168), it finds 241,580 triplets.
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)) c = 0 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: c += 1 print(f"{c:2}: {n:2}+{m:2}+{p:2} = {n + m + p}")
strange_triplets()</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
REXX
<lang rexx>/*REXX pgm lists triplet strange primes (<HI) where the sum of the primes sum is a 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 the absolute value of HI. */ if tell>0 then say 'list of unique triplet strange primes that sum to a prime:' call genP /*build array of semaphores for primes.*/ finds= 0 /*The number of 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 a prime? No, then skip it*/ finds= finds + 1 /*bump # of triple "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
<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.