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Problems & Puzzles:
Puzzles
Puzzle 1242 Recursive "Squarev" primes
Now we will work a
curio by G. L. Honaker, Jr.
Starting with the prime p1=89 square it and reverse
the result to obtain a new prime p2:
Process 1: p1 =89, 89^2
= 7921, rev(7921) = p2=1297
Repeat the process starting with the prime obtained
in the previos step:
Process 2: p2=1297,
1279^2
= 1682209. rev(1682209) = p3 = 9022861
Result: 89 (prime p1) -> 1297
(prime p2) -> 9022861(prime p3) -> 12352602021418
(Composite)
This
puzzle challenges you to start
with a prime p1 and get more than two new primes,
repeating the above process until a composite is
obtained in the last step.
Q,. Send
your largest sequence of primes.
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From Oct 18-25, 2025, contributions came from J. M. Rebert, Michael
Branicky, Giorgos kalogeropoulos, Gennady Gusev, Shyam Sunder Gupta,
Mike
Keith. Simon Cavegn, Adam Stinchcombe, Emmanuel Vantieghem.
***
Rebert wrote:
I found the following prime numbers: 10937,
30928211, 28028838001 which yield 3, 4, and 5 new prime numbers
respectively.
89 (prime p1) -> 1297 (prime p2) -> 9022861 (prime p3) ->
12352602021418 = 2 * 29 * 2693 * 79084997 (composite)
10937 (prime p1) ->
969716911 (prime p2) -> 129183974788053049 (prime p3) ->
10469183006010360852304024399488661 (prime p4) ->
129275762819439487280026443794964555233427340837353531633318297306901 =
7 * 71 * 260112198831870195734459645462705342521986601282401472099231986533
(composite)
30928211 (prime p1) -> 125066532455659 (prime p2) ->
18242110861060428404573614651 (prime p3) ->
108158148577729070474169051093721477136448812766806477233 (prime p4) ->
98263374338481976978647567285301592807839374095678223339516090076563466975963886957318619066205961126730158189611
(prime p5)
->
123133820310828501739569827171099568211081809875373474306329466105012110106708174826619974976668323496206407469279889340
3825072268079815048492889826065750399536448906599792679077099289959316591125606750254976038364836370965569
(composite)
28028838001 (prime p1) -> 100676103686957516587 (prime p2)
-> 96582108387371507648946802078535877653101 (prime p3)
->
1026194965309486474472028603842313781217741583204065247817943085277799450663018239
(prime p4)
->
1211666425820397074339256248641844689346521267068323492552434821784311306541660971136214903445842626137173291917796583508
294515279409701545733778418356286016703501 (prime p5)
->
1007565498753933821833082649276301569384670634609674826663160373214940929719315857966501311510565174677122719603439940224
189063502837021116915000425834974490082706669269367351505576274580296807411475121137173544905849163580547551759322410335982
582297788007172492133117179968747516356350583621695841703623695085730647255318641
(prime p6)
->
188680244260963336368214440727153970989170811578970899316797321166601549522902044021320655874120038511803585010013947140780
96085228666309901473744336502380663152647670779898812028563061469473310850943903772120622180175851825070889343694254633422982
31495892532793680582952522748658996032991664754558991345590424590405929962434778165523295829576267693868528094905272916071819
48125632695012707143053158891384365769045618887111288032132055937996791461380311998958599269143699977373170046192457747031376
26083262506947522427126070804267074792278542907032368547134117542913546065939103930424427651431239003178479909895327601309429
52007787143629724328815101 (composite)
***
Michael wrote:
I found the following record chains, given by length: prime1, ...,
primeN, composite
1: 2, 4
2: 19, 163, 96562
3: 89, 1297, 9022861,
12352602021418
4: 10937, 969716911, 129183974788053049,
10469183006010360852304024399488661,
129275762819439487280026443794964555233427340837353531633318297306901
5: 30928211, 125066532455659, 18242110861060428404573614651,
108158148577729070474169051093721477136448812766806477233,
9826337433848197697864756728530159280783937409567822333951609007656346697596388
6957318619066205961126730158189611,
123133820310828501739569827171099568211081809875373474306329466105012110106708174826619974976668323496
2064074692798893403825072268079815048492889826065750399536448906599792679077099289959316591125606750254976038364836370965569
6: 28028838001, 100676103686957516587,
96582108387371507648946802078535877653101,
10261949653094864744720286038423137812177415832040652478
17943085277799450663018239,
12116664258203970743392562486418446893465212670683234925524348217843113065416609711362149034458426261371732919
17796583508294515279409701545733778418356286016703501,
10075654987539338218330826492763015693846706346096748266631603732149409297193158579
665013115105651746771227196034399402241890635028370211169150004258349744900827066692693673515055762745802968074114751211371735449058491635
80547551759322410335982582297788007172492133117179968747516356350583621695841703623695085730647255318641,
188680244260963336368214440727153970989170811578970899316797321166601549522902044021320655874120038511803585010013947140780960852286663099
014737443365023806631526476707798988120285630614694733108509439037721206221801758518250708893436942546334229823149589253279368058295252274
865899603299166475455899134559042459040592996243477816552329582957626769386852809490527291607181948125632695012707143053158891384365769045
618887111288032132055937996791461380311998958599269143699977373170046192457747031376260832625069475224271260708042670747922785429070323685
4713411754291354606593910393042442765143123900317847990989532760130942952007787143629724328815101
***
Giorgos wrote: Here is my largest sequence with 6 primes:
{28028838001,
100676103686957516587,
96582108387371507648946802078535877653101,
1026194965309486474472028603842313781217741583204065247817943085277799450663018239,
121166642582039707433925624864184468934652126706832349255243482178431130654166097113621490344584262613717329191779658350829451527940970154573377
8418356286016703501,
100756549875393382183308264927630156938467063460967482666316037321494092971931585796650131151056517467712271960343994022418906350283
70211169150004258349744900827066692693673515055762745802968074114751211371735449058491635805475517593224103359825822977880071724921331171799687475163563
50583621695841703623695085730647255318641}
next term has 649 digits and
it is composite.
Later he wrote again:
Now I see that the
number I sent you has already been submitted to prime curios by submitter
Merickel but it has not been approved
yet.https://t5k.org/curios/page.php?short=28028838001
***
Gennady wrote:
Length=4: p1 = 10937, p2 = 969716911, p3 =
129183974788053049, p4 = 10469183006010360852304024399488661
Length=5: p1
= 30928211, p2 = 125066532455659, p3 = 18242110861060428404573614651, p4 =
108158148577729070474169051093721477136448812766806477233,
p5 =
98263374338481976978647567285301592807839374095678223339516090076563466975963886957318619066205961126730158189611
***
Shyam wrote:
Q. Send your largest sequence of primes.
The largest sequence of four more primes is obtained from the smallest
prime 30928211 (prime p1). There are not more than four more primes
starting from any prime up to p1 < 22801763489.30928211 -> 125066532455659
->18242110861060428404573614651 ->
108158148577729070474169051093721477136448812766806477233 ->
98263374338481976978647567285301592807839374095678223339516090076563466975963886957318619066205961126730158189611
There are large number of examples for one, two, three, and four more
primes.
Two smallest prime examples for one, two, three, and four
more primes are given below:19 -> 16337 -> 963189 -> 1297 -> 9022861313 ->
96979 ->144629404910937 ->969716911 ->129183974788053049 ->
1046918300601036085230402439948866156489 ->1217001913 ->
9659565426563901841 -> 1829813828183860820394568600324027033930928211
->125066532455659 -> 18242110861060428404573614651 ->
108158148577729070474169051093721477136448812766806477233 ->
98263374338481976978647567285301592807839374095678223339516090076563466975963886957318619066205961126730158189611103828861
->
12375567323408701 -> 104705288121018991122675666451351
->10252794235252156925361451615604511158149938525600650506379136901 ->
108388996106885100926535132205548224720464331848797488884175958928347973715461327133406035567674014649510683663168914036987911501
***
Mike Keith wrote:
The smallest prime that generates 3
more primes is 10937. The primes generated are
10937
969716911
129183974788053049
10469183006010360852304024399488661
The smallest
that generates 4 more primes is 30928211:
30928211
125066532455659
18242110861060428404573614651
108158148577729070474169051093721477136448812766806477233
98263374338481976978647567285301592807839374095678223339516090076563466975963886957318619066205961126730158189611
I think it will be quite hard to find one that generates 5 primes, but I
could be wrong. :-)
***
On Oct 22, 2025 Simon Cavegen sent the
record or champion for this search:
The first occurrence of seven
primes:
p1 = 19056157602311,
p2 =127045234023511465241731363,
p3 = 96773875814678591062016725144009170259788088419404161,
p4 =
1293114626818676434980294211135009701860150881052906068380164200036054048704878434564296393384910403815639,
p5 =
12387900305028227754386058490204021344229447777625686819243483158055869991409482027743631249639996480010428
95949606955687578495972176774408302805561286070003912595462343021037010765957304439370284042908345412761,
p6 =
12134616415970880349612987443352348021245269504531198717572232380198855250637153610564767880390416428904344
02172287998219087537808590870430764227217660582102974371641127612037424862482430288741019114253463612661100
52141885436516553730729325333181454034688408441285519418922599769642650196978757797382045646624485353038387
1501316416282804609969119568298341303585039436065153534491441986051012329438733932854813769370064351,
p7 =
10215018816454683403837191992521460999060481575486835124991488600090718774821769434394638777727116266806331
51378174218542439764142401695173426698653530200681704841669909542866227900830591565968183855438122125086990
75123375995566928717318168617140189623963927054509165023674618259319250769317073188480782866204229419363139
94824469954399769899634244651121559767462403134864497042863242351047328333681188330277315553594580655684037
96835992220036037531916716628304092099355827206880494599935100708775563851661267253241624441713787771309594
94232370574477703300210534843217470957008401470389187352180322481036730635469844014665932653339927419441120
10618383895919322180350298546406222292786395080828131570669918420907461346810070009875843081071818786575952
67170172490577915013232354945871773290381649005670630285602948703100277084379947265519842741...
NB by CR: 1) All primes verified by CR using the web online Integer
Factorizatins Calculator by Darío Alpern.
2) Prime p7 = 841 digits. 3)
Simon has the champion or the Record for this puzzle, just by now...
***
Adam Stinchcombe wrote
I found 50 different starting p's
that led to sequences of length 4, then I cut that off:[3116749, 8584817,
10453567, 10824391, 12232991,
12621241, 27590791, 30738041, 95173151,
102646337, 109470833, 119840737, 128721991, 129575917, 176330467, 190427113,
190431863, 197393051,
270401039, 277341391, 301293667, 307172941,
313471441, 315003401, 333493763, 336545717, 340321337, 342445109, 351012533,
374533099,
379233037, 381685783, 383150087, 383616539, 384474611,
392544391, 396977941, 401010559, 401457059, 405550133, 440901089, 600383989,
603153011, 621955309, 623121691, 848351809, 860066401, 873402641, 949172617,
976938793]
I found only a few p's that led to sequences of length 5:
124970887, 988265749, 315497089
and even fewer p's that led to sequences
of length 6: 28028838001, 99174318917
So far I have found of longer
length. I was picking random starting p values, so these are not exhaustive
nor necessarily minimal.
***
Emmanuel wrote:
Here is my
largest sequence :
28028838001
-> 100676103686957516587
->
96582108387371507648946802078535877653101
->
1026194965309486474472028603842313781217741583204065247817943085277799450663018239
->
,12116664258203970743392562486418446893465212670683234925524348217843113065416609711362149034458426261371732919177
96583508294515279409701545733778418356286016703501
->100756549875393382183308264927630156938467063460967482666316037321494092971931585796650131151056517467712271960343
99402241890635028370211169150004258349744900827066692693673515055762745802968074114751211371735449058491635805475517
59322410335982582297788007172492133117179968747516356350583621695841703623695085730647255318641(six
primes)
The next step leads to a compoiste number
18868024426096333636821444072715397098917081157897089931679732116660154952290204402132065587412003851180358501001394
71407809608522866630990147374433650238066315264767077989881202856306146947331085094390377212062218017585182507088934
36942546334229823149589253279368058295252274865899603299166475455899134559042459040592996243477816552329582957626769
38685280949052729160718194812563269501270714305315889138436576904561888711128803213205593799679146138031199895859926
91436999773731700461924577470313762608326250694752242712607080426707479227854290703236854713411754291354606593910393
042442765143123900317847990989532760130942952007787143629724328815101 (649
digits, divisible by 157)
***
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