Puzzle 1224 Primes and Factoriangular integers

Puzzle 1224 Primes and Factoriangular integers

Oscar Volpati sent the following puzzle:

Factoriangular integers:

Link 1, 2

3, 7, 13 & 17 are primes that are the sum of two factoriangular integers.

3 = 1+2 = ft(0)+ft(1),
13 = 1+12 = ft(0)+ft(3).
7 = 2+5 = ft(1)+ft(2),
17 = 5+12 = ft(2)+ft(3)

Q. Which primes are the sum of two factoriangular integers (send ten more of these)?

From Jun 7-13, 2025, contributions came from Giorgos Kalogeropoulos, Paul Cleary, Emmanuel Vantieghem, Gennady Gusev, Oscar Volpatti, Morné Louw, Simon Cavegn

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Giorgos wrote:

I searched the first 1000 factoriangular numbers and found 183 pairs.

Here are the indexes of the pairs, i.e. {0,1} means ft(0)+ft(1)

{0, 1}, {0, 3}, {0, 8}, {0, 20}, {0, 43}, {0, 135}, {0, 288}, {1, 2}, {1, 5}, {1, 6}, {1, 9}, {1, 18}, {1, 914}, {2, 3}, {2, 8}, {2, 12}, {3, 109}, {6, 244}, {7, 10}, {7, 29}, {7, 50}, {7, 422}, {9, 52}, {10, 147}, {13, 19}, {13, 23}, {13, 59}, {13, 283}, {13, 656}, {16, 22}, {16, 26}, {16, 34}, {16, 42}, {16, 46}, {16, 101}, {17, 55}, {19, 26}, {19, 46}, {20, 46}, {22, 84}, {29, 91}, {31, 269}, {33, 280}, {36, 130}, {37, 47}, {37, 948}, {40, 282}, {43, 446}, {43, 929}, {44, 202}, {46, 56}, {46, 127}, {46, 239}, {46, 547}, {47, 121}, {51, 349}, {52, 970}, {55, 78}, {55, 877}, {57, 883}, {57, 1000}, {58, 96}, {58, 163}, {61, 267}, {66, 112}, {67, 90}, {67, 209}, {67, 250}, {67, 286}, {74, 151}, {74, 307}, {75, 118}, {79, 598}, {82, 740}, {82, 891}, {82, 971}, {84, 757}, {88, 133}, {91, 717}, {93, 100}, {94, 371}, {95, 142}, {98, 292}, {106, 152}, {106, 155}, {106, 191}, {111, 733}, {118, 863}, {119, 166}, {130, 287}, {137, 715}, {154, 171}, {157, 184}, {159, 313}, {160, 186}, {160, 358}, {161, 799}, {172, 378}, {184, 633}, {184, 717}, {186, 787}, {187, 217}, {187, 389}, {187, 862}, {188, 670}, {189, 796}, {190, 628}, {191, 205}, {191, 250}, {193, 444}, {201, 712}, {202, 227}, {207, 370}, {208, 466}, {217, 243}, {222, 844}, {227, 790}, {229, 492}, {233, 328}, {247, 750}, {256, 489}, {259, 313}, {260, 877}, {268, 641}, {269, 352}, {274, 371}, {277, 316}, {287, 850}, {296, 334}, {300, 373}, {304, 542}, {316, 461}, {325, 583}, {325, 627}, {327, 565}, {328, 445}, {331, 881}, {337, 443}, {341, 559}, {343, 438}, {345, 388}, {348, 349}, {361, 899}, {367, 374}, {367, 497}, {367, 525}, {367, 822}, {381, 436}, {383, 478}, {386, 496}, {400, 618}, {422, 928}, {424, 513}, {439, 698}, {443, 874}, {445, 543}, {448, 877}, {484, 618}, {485, 967}, {490, 816}, {490, 943}, {508, 778}, {513, 883}, {517, 584}, {525, 991}, {532, 542}, {538, 815}, {556, 829}, {559, 841}, {568, 806}, {601, 632}, {611, 733}, {631, 817}, {643, 697}, {703, 938}, {746, 943}, {751, 881}, {767,
874}, {771, 841}, {775, 778}, {818, 919}, {848, 874}, {952, 989}

The biggest prime from the pair {952, 989} is 425220761610961330056......000943183 (2535 digits) but I guess even bigger primes can be found easily

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Paul wrote:

Here are the ft indexes of 104 solutions (100 + the 4 supplied). There is a huge amount of data so only the indexes supplied.

{0,1}
{0,3}
{0,8}
{0,20}
{0,43}
{0,135}
{0,288}
{1,2}
{1,5}
{1,6}
{1,9}
{1,18}
{2,3}
{2,8}
{2,12}
{3,109}
{6,244}
{7,10}
{7,29}
{7,50}
{7,422}
{9,52}
{10,147}
{13,19}
{13,23}
{13,59}
{13,283}
{16,22}
{16,26}
{16,34}
{16,42}
{16,46}
{16,101}
{17,55}
{19,26}
{19,46}
{20,46}
{22,84}
{29,91}
{31,269}
{33,280}
{36,130}
{37,47}
{40,282}
{43,446}
{44,202}
{46,56}
{46,127}
{46,239}
{47,121}
{51,349}
{55,78}
{58,96}
{58,163}
{61,267}
{66,112}
{67,90}
{67,209}
{67,250}
{67,286}
{74,151}
{74,307}
{75,118}
{88,133}
{93,100}
{94,371}
{95,142}
{98,292}
{106,152}
{106,155}
{106,191}
{119,166}
{130,287}
{154,171}
{157,184}
{159,313}
{160,186}
{160,358}
{172,378}
{187,217}
{187,389}
{191,205}
{191,250}
{193,444}
{202,227}
{207,370}
{208,466}
{217,243}
{233,328}
{259,313}
{269,352}
{274,371}
{277,316}
{296,334}
{300,373}
{316,461}
{328,445}
{337,443}
{343,438}
{345,388}
{348,349}
{367,374}
{381,436}
{383,478}

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Emmanuel wrote:

Here are the next 10 :
137 = ft(2) + ft(6) = 2 + 135
743 = ft(2) + ft(7) = 2 + 741
40357 = ft(1) + ft(9) = 1 + 40356
40361 = ft(3) + ft(9) = 5 + 40356
362927 = ft(2) + ft(10) = 2 + 362925
3633923 = ft(8) + ft(11) = 5068 + 3628855
479001683 = ft(3) + ft(13) = 5 + 479001678
6402373705728173 = ft(2) + ft(19) = 2 + 6402373705728171
121645106635853081 = ft(14) + ft(20) = 6227020891 + 121645100408832190
2432902008176640211 = ft(1) + ft(21) = 1 + 2432902008176640210

I easily found many more such primes .

But they soon become so big that primality proving will be needed.

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Gennady wrote:

There were found 37 primes for n<=100:

ft(0)+ft(0)=2

ft(0)+ft(1)=3

ft(1)+ft(2)=7

ft(0)+ft(3)=13

ft(2)+ft(3)=17

ft(1)+ft(5)=137

ft(1)+ft(6)=743

ft(0)+ft(8)=40357

ft(2)+ft(8)=40361

ft(1)+ft(9)=362927

ft(7)+ft(10)=3633923

ft(2)+ft(12)=479001683

ft(1)+ft(18)=6402373705728173

ft(13)+ft(19)=121645106635853081

ft(0)+ft(20)=2432902008176640211

ft(16)+ft(22)=1124000748700397568389

ft(13)+ft(23)=25852016738891203661167

ft(16)+ft(26)=403291461126626558373888487

ft(19)+ft(26)=403291461248250735992832541

ft(7)+ft(29)=8841761993739701954543616005503

ft(16)+ft(34)=295232799039604140847618630566309888731

ft(16)+ft(42)=1405006117752879898543142606244511569957306789889039

ft(0)+ft(43)=60415263063373835637355132068513997507264512000000947

ft(16)+ft(46)=5502622159812088949850305428800254892961651773882789889217

ft(19)+ft(46)=5502622159812088949850305428800254892961773398060408833271

ft(20)+ft(46)=5502622159812088949850305428800254892964084654968176641291

ft(37)+ft(47)=258623241511168194406717446379957026285177213970022400001831

ft(7)+ft(50)=30414093201713378043612608166064768844377641568960512000000006343

ft(9)+ft(52)=80658175170943878571660636856403766975289505440883277824000000364303

ft(17)+ft(55)=12696403353658275925965100847566516959580321051449436762276195687428097693

ft(46)+ft(56)=710998587804863457356667807275813899586803407681423351649098792960000002677

ft(13)+ft(59)=138683118545689835737939019720389406345902876772687432540821294940160006227022661

ft(55)+ft(78)=113242811782062978314575211587320462287317622758916056483248887907696828917511603
46566137662614106275840000000004621

ft(22)+ft(84)=331424013456535326699938757913013128800066628624204948711884603238305913129171686
4129885722968716753156179044000727777607683823

ft(67)+ft(90)=148571596448176149730952273362082573788556999775579968513090215195484448969953030
3713627445420408653347245734737940548943872000
000000006373

ft(29)+ft(91)=135200152767840296255166568759495142147586866476906677791741734597153670771559994
765685283954750449427751168345609770185739701954543616004621

ft(58)+ft(96)=99167793487094968920957140154189380115818364865126779544437605483849222515965283127056804786
6475659497149778567828072486782798980699914240000000006367

and a large prime with 10000+ digits:

ft(203)+ft(3253) - (10015 digits) is prime.

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Oscar wrote:

The smallest solution is actually prime 2:

ft(0)+ft(0) = 2.

Any further solution must be the sum of distinc factoriangular numbers.

After prime 17, next ten solutions are:

ft(5)+ft(1) = 137,

ft(6)+ft(1) = 743,

ft(8)+ft(0) = 40357,

ft(8)+ft(2) = 40361,

ft(9)+ft(1) = 362927,

ft(10)+ft(7) = 3633923,

ft(12)+ft(2) = 479001683,

ft(18)+ft(1) = 6402373705728173,

ft(19)+ft(13) = 121645106635853081,

ft(20)+ft(0) = 2432902008176640211.

There are no further solutions below 2^64.

Factoriangular numbers grow very quickly, which makes primality test increasingly difficult.

Nevertheless, solutions seem to be quite common, so that it can be conjectured that there are infinitely many solutions.

A more precise counting conjecture can be formulated:

let FT2(x) be the number of primes of the form ft(n)+ft(m),

with indexes 0 <= m <= n <= x;

then FT2(x) grows like x/log(x), as fast as the number of all primes not exceeding x.

Attached figure P1224OV.jpg allows to compare FT2(x), blue line, with PrimePi(x), red line, for x < 1200.

(Maybe justification /explanation of such conjecture could be a nice followup puzzle?)

Two solutions catched my interest because both indexes n and m are prime too:

ft(3)+ft(2) = 17,

ft(19)+ft(13) = 121645106635853081.

I searched for more solutions of this special form, where prime indexes n and m allow to generate a solution with d digits:

n m d

3 2 2

19 13 18

23 13 23

29 7 31

47 37 60

59 13 81

109 3 177

269 31 539

283 13 573

443 337 982

881 331 2214

881 751 2214

929 43 2356

1033 547 2667

1033 691 2667

1783 113 5025

2713 2399 8139

3571 2477 11139

4639 2801 14997

Solutions with less than 1000 digits were proven prime using APRCL;

remaining solutions are PRPs passing BPSW test.

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Morné wrote:

137 = ft(1) + ft(5)

743 = ft(1) + ft(6)

40357 = ft(0) + ft(8)

40361 = ft(2) + ft(8)

362927 = ft(1) + ft(9)

3633923 = ft(7) + ft(10)

479001683 = ft(2) + ft(12)

6402373705728173 = ft(1) + ft(18)

121645106635853081 = ft(13) + ft(19)

2432902008176640211 = ft(0) + ft(20)

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Simon wrote:
3
7
13
17
137
743
40357
40361
362927
3633923
479001683
6402373705728173
121645106635853081
2432902008176640211

See Pu1224 SC.txt for first 430 results.

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