Find 7 digit prime numbers with this property;

Question

When you subtract the sum of the squares of the digits of the number from the original number it gives you another prime number squared.

Answer 1

EDIT:

Okay, here's my solution:

for i in `primes 10000 100000`;do x=$(echo $i|python -c 'import sys, math;s = sys.stdin.read()[:-1]; p = math.sqrt(int(s) - sum(map(lambda x : int(x) * int(x), list(s))));                                                                                
if p % 1 == 0: print str(int(p))')
    if [ "$x" != "" ];then
        if  [ $(factor $x|awk '{ print NF }') -eq "2" ];then
                    echo $i
    fi
    fi
done

Ugly code, but I get:

18973
24763
37321
73561
96953

Let's check 37321. We get 37321 - 3² - 7² - 3² - 2² - 1², whose square root is 193, which is prime.

For 7 digits, here's what I have so far:

1100509
…

Let's check 1100509. We get sqrt(1100509 - 1 - 1 - 25 - 81) = 1049, which is prime. Feel free to expend more CPU cycles to get more.

Answer 2

My previous solution was not checking that the numbers are prime, just that $n - \!\!\!\!\!\!\!\!\sum\limits_{\text{digits $d$ of $n$}} d^2$ are.

So, in Mathematica:

n = 7;
Select[Range[10^(n - 1) + 1, 10^n - 1, 2],
  And[
    PrimeQ[#1],
    PrimeQ[Sqrt[#1 - Apply[Plus, IntegerDigits[#1]^2]]]
  ] &
]

Output:

{1100509, 1585201, 1661789, 2211257, 4541309, 4871077, 4897709,
5340949, 5958751, 7393123, 8185501, 8744003}

Answer 3

The primes from $1000$ to $10^8$ with that property (and the corresponding root) are:

(2903,53)
(3533,59)
(3803,61)
(5197,71)
(9533,97)
(18973,137)
(24763,157)
(37321,193)
(73561,271)
(96953,311)
(113621,337)
(124777,353)
(129097,359)
(134837,367)
(139241,373)
(398341,631)
(830003,911)
(1100509,1049)
(1585201,1259)
(1661789,1289)
(2211257,1487)
(4541309,2131)
(4871077,2207)
(4897709,2213)
(5340949,2311)
(5958751,2441)
(7393123,2719)
(8185501,2861)
(8744003,2957)
(11485559,3389)
(15343039,3917)
(15343079,3917)
(16008193,4001)
(16459441,4057)
(16736519,4091)
(17648693,4201)
(17901563,4231)
(21077401,4591)
(22915511,4787)
(22915591,4787)
(25877861,5087)
(26553629,5153)
(28058447,5297)
(28058467,5297)
(30261149,5501)
(32137747,5669)
(33443269,5783)
(34304561,5857)
(36954461,6079)
(40360753,6353)
(43073159,6563)
(43309751,6581)
(47375963,6883)
(47596441,6899)
(48762557,6983)
(53743763,7331)
(56505481,7517)
(62141909,7883)
(65496941,8093)
(67782521,8233)
(76510171,8747)
(80982263,8999)
(82864927,9103)
(83302253,9127)
(84401197,9187)
(85027057,9221)
(86360003,9293)
(90992939,9539)
(95394619,9767)
(96373819,9817)

Code (Haskell):

module Main (main) where

import Math.NumberTheory.Primes
import Math.NumberTheory.Powers

import Control.Monad (guard)
import System.Environment (getArgs)

main :: IO ()
main = do
    args <- getArgs
    let (lo,hi) = case args of
                    a:b:_ -> (read a, read b)
                    a:_   -> (1000, read a)
                    _     -> (1000, 10^7)
    mapM_ print $ funPrimes lo hi

digitSquareSum :: Integer -> Integer
digitSquareSum = go 0
  where
    go acc 0 = acc
    go acc m = case m `quotRem` 10 of
                 (q,r) -> go (acc + r*r) q

rootPrime :: Integer -> Maybe Integer
rootPrime n = do
    let m = n - digitSquareSum n
    r <- exactSquareRoot m
    guard (isPrime r)
    return r

funPrimes :: Integer -> Integer -> [(Integer, Integer)]
funPrimes low high = takeWhile (<= high) (sieveFrom low) >>= \p ->
                                    case rootPrime p of
                                      Just r -> [(p,r)]
                                      Nothing -> []

Between $10^8$ and $10^9$, we have

(105411479,10267)
(110271107,10501)
(111492631,10559)
(111492671,10559)
(116014531,10771)
(116014571,10771)
(117787913,10853)
(117787993,10853)
(117961573,10861)
(120846227,10993)
(134258869,11587)
(148767061,12197)
(149793487,12239)
(156325123,12503)
(161010841,12689)
(161366489,12703)
(165199973,12853)
(166900837,12919)
(167780441,12953)
(169078339,13003)
(169234399,13009)
(171846139,13109)
(184878781,13597)
(187334143,13687)
(188485741,13729)
(199007723,14107)
(207043433,14389)
(207677111,14411)
(208023083,14423)
(209641673,14479)
(214300417,14639)
(218182633,14771)
(221384893,14879)
(235039859,15331)
(235899239,15359)
(238486597,15443)
(242082611,15559)
(247338829,15727)
(258020197,16063)
(258791927,16087)
(258791987,16087)
(262083953,16189)
(267944477,16369)
(270701371,16453)
(275792723,16607)
(276191479,16619)
(286320427,16921)
(286320487,16921)
(288354637,16981)
(288762337,16993)
(312193733,17669)
(312688781,17683)
(316519933,17791)
(319730353,17881)
(328298413,18119)
(343694849,18539)
(357928927,18919)
(359443943,18959)
(365612881,19121)
(368909131,19207)
(371217443,19267)
(371217463,19267)
(393744983,19843)
(399720367,19993)
(417916507,20443)
(424319009,20599)
(441798719,21019)
(454414673,21317)
(462981581,21517)
(472497401,21737)
(475371007,21803)
(486776309,22063)
(486952877,22067)
(487482577,22079)
(490932931,22157)
(500282917,22367)
(503598839,22441)
(508187159,22543)
(508457681,22549)
(511619333,22619)
(511709977,22621)
(520615621,22817)
(520615681,22817)
(524730821,22907)
(532271219,23071)
(562591219,23719)
(565155733,23773)
(575088641,23981)
(575088799,23981)
(622153489,24943)
(648364667,25463)
(658281941,25657)
(671794927,25919)
(671794987,25919)
(702939491,26513)
(726464467,26953)
(739350793,27191)
(744362309,27283)
(785737283,28031)
(808435721,28433)
(811851211,28493)
(863360941,29383)
(864419033,29401)
(878589319,29641)
(940097269,30661)
(949194871,30809)
(959946707,30983)
(968641477,31123)
(974876041,31223)
(978251059,31277)
(990801919,31477)

$111$ more. So these beasts are rare, but there are enough.