DigitSums of some consecutive primes

 
 
(1) Find four consecutive primes having digit sums that, in some order, are consecutive primes

(2) Find five consecutive primes having digit sums that are distinct primes

 
 
Starting with   (2,   3,   5,   7)   and   (3,   5,   7,   11)

 
Conjecture :   The problem has infinitely many solutions.

 
 
 
 
Paul’s solution:

{191,193,197,199}   ,   {11,13,17,19}
{821,823,827,829}   ,   {11,13,17,19}
{1321,1327,1361,1367}   ,   {7,11,13,17}
{2081,2083,2087,2089}   ,   {11,13,17,19}
{2351,2357,2371,2377}   ,   {11,13,17,19}
{2551,2557,2579,2591}   ,   {13,17,19,23}
{3251,3253,3257,3259}   ,   {11,13,17,19}
{3257,3259,3271,3299}   ,   {13,17,19,23}
{5897,5903,5923,5927}   ,   {17,19,23,29}
{8597,8599,8609,8623}   ,   {19,23,29,31}
{10079,10091,10093,10099}   ,   {11,13,17,19}
{10433,10453,10457,10459}   ,   {11,13,17,19}
{11171,11173,11177,11197}   ,   {11,13,17,19}
{12301,12323,12329,12343}   ,   {7,11,13,17}
{12451,12457,12473,12479}   ,   {13,17,19,23}
{13001,13003,13007,13009}   ,   {5,7,11,13}
{14327,14341,14347,14369}   ,   {13,17,19,23}
{17597,17599,17609,17623}   ,   {19,23,29,31}
{21011,21013,21017,21019}   ,   {5,7,11,13}
{24023,24029,24043,24049}   ,   {11,13,17,19}
{25031,25033,25037,25057}   ,   {11,13,17,19}
{25301,25303,25307,25309}   ,   {11,13,17,19}
{27077,27091,27103,27107}   ,   {13,17,19,23}
{31333,31337,31357,31379}   ,   {13,17,19,23}
{34019,34031,34033,34039}   ,   {11,13,17,19}
{34183,34211,34213,34217}   ,   {11,13,17,19}
{35171,35201,35221,35227}   ,   {11,13,17,19}
{36011,36013,36017,36037}   ,   {11,13,17,19}
{39799,39821,39827,39829}   ,   {23,29,31,37}
{44021,44027,44029,44041}   ,   {11,13,17,19}
{45491,45497,45503,45523}   ,   {17,19,23,29}
{58897,58901,58907,58909}   ,   {23,29,31,37}
{65899,65921,65927,65929}   ,   {23,29,31,37}
{90107,90121,90127,90149}   ,   {13,17,19,23}
{92893,92899,92921,92927}   ,   {23,29,31,37}
{101111,101113,101117,101119}   ,   {5,7,11,13}
{101207,101209,101221,101267}   ,   {7,11,13,17}
{102251,102253,102259,102293}   ,   {11,13,17,19}
{102253,102259,102293,102299}   ,   {13,17,19,23}
{104123,104147,104149,104161}   ,   {11,13,17,19}
{104323,104327,104347,104369}   ,   {13,17,19,23}

 
Part 2
Same prime range

{1291,1297,1301,1303,1307}   ,   {13,19,5,7,11}
{3257,3259,3271,3299,3301}   ,   {17,19,13,23,7}
{10079,10091,10093,10099,10103}   ,   {17,11,13,19,5}
{12983,13001,13003,13007,13009}   ,   {23,5,7,11,13}
{33997,34019,34031,34033,34039}   ,   {31,17,11,13,19}
{35159,35171,35201,35221,35227}   ,   {23,17,11,13,19}
{35759,35771,35797,35801,35803}   ,   {29,23,31,17,19}
{42193,42197,42209,42221,42223}   ,   {19,23,17,11,13}
{54983,55001,55009,55021,55049}   ,   {29,11,19,13,23}
{67181,67187,67189,67211,67213}   ,   {23,29,31,17,19}
{67187,67189,67211,67213,67217}   ,   {29,31,17,19,23}
{102071,102077,102079,102101,102103}   ,   {11,17,19,5,7}
{102077,102079,102101,102103,102107}   ,   {17,19,5,7,11}
{102251,102253,102259,102293,102299}   ,   {11,13,19,17,23}
{102253,102259,102293,102299,102301}   ,   {13,19,17,23,7}
 
 
 
 
 
 
 
 
 
 
 
 

Advertisements

About benvitalis

math grad - Interest: Number theory
This entry was posted in Prime Numbers and tagged . Bookmark the permalink.

4 Responses to DigitSums of some consecutive primes

  1. paul says:

    Part 1.
    Primes <=10000.

    {191,193,197,199} , {11,13,17,19}
    {821,823,827,829} , {11,13,17,19}
    {1321,1327,1361,1367} , {7,11,13,17}
    {2081,2083,2087,2089} , {11,13,17,19}
    {2351,2357,2371,2377} , {11,13,17,19}
    {2551,2557,2579,2591} , {13,17,19,23}
    {3251,3253,3257,3259} , {11,13,17,19}
    {3257,3259,3271,3299} , {13,17,19,23}
    {5897,5903,5923,5927} , {17,19,23,29}
    {8597,8599,8609,8623} , {19,23,29,31}
    {10079,10091,10093,10099} , {11,13,17,19}
    {10433,10453,10457,10459} , {11,13,17,19}
    {11171,11173,11177,11197} , {11,13,17,19}
    {12301,12323,12329,12343} , {7,11,13,17}
    {12451,12457,12473,12479} , {13,17,19,23}
    {13001,13003,13007,13009} , {5,7,11,13}
    {14327,14341,14347,14369} , {13,17,19,23}
    {17597,17599,17609,17623} , {19,23,29,31}
    {21011,21013,21017,21019} , {5,7,11,13}
    {24023,24029,24043,24049} , {11,13,17,19}
    {25031,25033,25037,25057} , {11,13,17,19}
    {25301,25303,25307,25309} , {11,13,17,19}
    {27077,27091,27103,27107} , {13,17,19,23}
    {31333,31337,31357,31379} , {13,17,19,23}
    {34019,34031,34033,34039} , {11,13,17,19}
    {34183,34211,34213,34217} , {11,13,17,19}
    {35171,35201,35221,35227} , {11,13,17,19}
    {36011,36013,36017,36037} , {11,13,17,19}
    {39799,39821,39827,39829} , {23,29,31,37}
    {44021,44027,44029,44041} , {11,13,17,19}
    {45491,45497,45503,45523} , {17,19,23,29}
    {58897,58901,58907,58909} , {23,29,31,37}
    {65899,65921,65927,65929} , {23,29,31,37}
    {90107,90121,90127,90149} , {13,17,19,23}
    {92893,92899,92921,92927} , {23,29,31,37}
    {101111,101113,101117,101119} , {5,7,11,13}
    {101207,101209,101221,101267} , {7,11,13,17}
    {102251,102253,102259,102293} , {11,13,17,19}
    {102253,102259,102293,102299} , {13,17,19,23}
    {104123,104147,104149,104161} , {11,13,17,19}
    {104323,104327,104347,104369} , {13,17,19,23}

    Paul.

    • paul says:

      Part 2
      Same prime range

      {1291,1297,1301,1303,1307} , {13,19,5,7,11}
      {3257,3259,3271,3299,3301} , {17,19,13,23,7}
      {10079,10091,10093,10099,10103} , {17,11,13,19,5}
      {12983,13001,13003,13007,13009} , {23,5,7,11,13}
      {33997,34019,34031,34033,34039} , {31,17,11,13,19}
      {35159,35171,35201,35221,35227} , {23,17,11,13,19}
      {35759,35771,35797,35801,35803} , {29,23,31,17,19}
      {42193,42197,42209,42221,42223} , {19,23,17,11,13}
      {54983,55001,55009,55021,55049} , {29,11,19,13,23}
      {67181,67187,67189,67211,67213} , {23,29,31,17,19}
      {67187,67189,67211,67213,67217} , {29,31,17,19,23}
      {102071,102077,102079,102101,102103} , {11,17,19,5,7}
      {102077,102079,102101,102103,102107} , {17,19,5,7,11}
      {102251,102253,102259,102293,102299} , {11,13,19,17,23}
      {102253,102259,102293,102299,102301} , {13,19,17,23,7}

      Paul.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s