## 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}

math grad - Interest: Number theory
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### 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.

• benvitalis says:

It starts with (2,3,5,7) and (3,5,7,11)

• benvitalis says:

I’m gonna throw a conjecture: the problem has infinitely many solutions.
I have no idea how to prove it, though.