[Math] Find 7 digit prime numbers with this property;

prime numberspuzzlerecreational-mathematics

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.

Best Answer

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.

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