20
$\begingroup$

Title asks it: Does the Fourier expansion of the j-function have any prime coefficients?

A superabundance of congruences involving primes up to 13 rule out many candidates, but calculation suggests that primes $p>13$ occur as divisors at frequencies (about?) $1/p$.

But $$c_{71}=278775024890624328476718493296348769305198947=(353) (5533876049689057963) (142708463580969897033673)$$ so that might count as a near miss.

That said, though composite, none of $c_{1319},c_{1559},c_{1871},c_{2111},c_{2231},c_{3239},c_{3551}, c_{4271}, c_{4799}, c_{5471}$,... has a factor less than 100.

Improve this question
$\endgroup$
10
  • 7
    $\begingroup$ The coefficients $c_{n}$ for $1 \leq n \leq 5 \cdot 10^{5}$ are all composite. Given that $c_{n} \sim e^{4 \pi \sqrt{n}}/(\sqrt{2} n^{3/4})$ it seems likely that there are some prime coefficients. $\endgroup$
    – Jeremy Rouse
    Commented Nov 21, 2020 at 16:17
  • 2
    $\begingroup$ Tabulation of the coefficients with many links to the lliterature at oeis.org/A000521 Factorizations up to $n=1000$ given at asahi-net.or.jp/~KC2H-MSM/mathland/matha1 $\endgroup$
    – Gerry Myerson
    Commented Nov 21, 2020 at 23:20
  • $\begingroup$ @JeremyRouse In fact, given that growth rate, our naive expectation should be infinitely many prime coefficients. $\endgroup$
    – JoshuaZ
    Commented Nov 25, 2020 at 21:25
  • 1
    $\begingroup$ I discovered a bug in my program (a type error) that leads to all the coefficients being labelled "not prime". $\endgroup$
    – Jeremy Rouse
    Commented Nov 25, 2020 at 22:21
  • 1
    $\begingroup$ The sum over the primes $p>13$ of $1/p$ (your expected frequencies) is divergent, so maybe a study of the relative frequency $f_{k}$ of the coefficients having exactly $k$ prime factors counted with multiplicity could shed some light on the problem. $\endgroup$
    – Sylvain JULIEN
    Commented Nov 25, 2020 at 22:37

1 Answer 1

Reset to default
23
$\begingroup$

There are seven prime values (passing a BPSW test) of $c_n$ with $n \le 2 \cdot 10^7$, at indices 457871, 685031, 1029071, 1101431, 9407831, 11769911, and 18437999.

For a writeup about the computations, source code, and the prime numbers themselves, see: https://github.com/fredrik-johansson/jfunction

The first prime $c_{457871}$ is the following 3689-digit number:

30801636514011810817665318374658229108845878479020424564852145048550477094266397753615620563343543122809859646561082297428127434662591162587465688189470673559500946738365120052237063677089165938711865291826029106118944935275575747924692016273156986404534483801538929131770513483035012136870394657959401728135298020394188493171098233244180987908576898694823463564573986521378977434641833939268448907892425327423931702787985965542437137823321051154070271812702040044018249180573203850835848455588503282760349067467783417988439399728717693269653344946303351200847243267479008953480983893357261716684979093234554524907570654175153716092013381021033631140271230374127359907711968372522745844402375355449952404485703394467619855566837872874898909971592754384802113052943616244616228982238122927893868687205058321497136583843454449483568143398274104240118934930464297655670020151675237805373543868606164624359906981163728280265160187710553222515223577715624809615213830941718538338330718318212962253165154784990727034694055424785446779353587418545233599512401297307560287330719685869795522920270669609498639686158362603820371777436303198350821510562028940026062291473250204511985677264132923030166903832835499326969147438699601413777550018800825631136927531190121891919023537853918707520543367800297170322643557011053113196699442392677721251164071912545596204918917566579754931649583289900867072630307177448327512748573948720052532426026582228531694404885199310535255097152264726060203165225244982020696634699998363392968546865623955653923292400203238704016331998301098882719280237294694368706187727194309664172903851082095220551328392538250319846241916349485122880753951745944930504836900929002975743487891022913325212927877123283110371004511930228354316650588540693893785256904915901446318107624914828072059779823732136657855374843120155979700684426795681799949828354496446059153417627020927839115150227802126261918915561804876068184778844391781134858921611498880947743310539662060436623031296503497690935576519684398269951037522254928920040638913033916447480995451172728711833839757712418893132702856601768693707767408660072694546475350179623544013680656723320135856226549734491873157405584154845513711221581080660024268777736123563250061564832634510929167076111162819878757339574946474890631427413494266622011134541083389565676533915214975782362935488902596227893270340685189472412328774565836766141144309983845007818359559400419277334332676913295096744471767890050480297665094988593656013858404531156346968039028944208325870737014099138274367178622028282421548639862174878971583355345514824564041789417768026191489674685601130884770272042320774286085741557976163690129647685531361972350643878463985533720888119015787469601005630621659687749812528402893568495431951749454436584549713954945990865871059056586397539337257860037142247888078726810588665972975365062119719322236620399267595421045468328581519496916075539958247075033250213251987450634988759698630929366797978086037651833328039578740169593342409162117195011259429537049154618053581738781731898468529654264215451362063751162610494664217028387863630499363451607405876105244341222047086047630946033922722790116254686279821386138836326264599571591967081705529398847838399554521937548523334357827287180353919911626556542110462725508733594289737987833526697420630616509010365868192637613354631155273335447830149024265581051436057754643260886600218454986343035650276055047068090749733940363539512760186969985697586068479926077515927351409212093813617210155525035138051400826519032339949113881712130873854263099726139035786383205124051644483087596049043497124737908938233702928805190964560673396398573801431769834039050534841422792714076699

Improve this answer
$\endgroup$
13
  • 3
    $\begingroup$ FWIW: Mathematica confirms that the two numbers listed above are prime. $\endgroup$
    – Aeryk
    Commented Nov 25, 2020 at 19:31
  • 2
    $\begingroup$ @SylvainJULIEN Not sure what method Fredrik used, but Pari GP (and hence also Sage) use the Baillie-PSW primality test for pseudoprimality. No failures of this test are known, but I don't think any standard conjectures guarantee its correctness. According to the article Mathematica's PrimeQ also uses this test. $\endgroup$
    – Wojowu
    Commented Nov 25, 2020 at 19:59
  • 2
    $\begingroup$ I checked with Pari and Flint which both implement (possibly slightly different versions of) BPSW which has no known counterexamples. In any case, these numbers are small enough that they can be certified prime in reasonable time (hours or days). $\endgroup$
    – Fredrik Johansson
    Commented Nov 25, 2020 at 20:04
  • 4
    $\begingroup$ Fantastic. Thanks. Proving now that infinitely many such primes exist seems very difficult. $\endgroup$
    – David Feldman
    Commented Nov 25, 2020 at 20:45
  • 3
    $\begingroup$ The ECPP test completed - $c_{457871}$ is prime. Primo 4.3.2 took 11336 seconds to verify it (running on six cores on a fairly old machine). $\endgroup$
    – Jeremy Rouse
    Commented Nov 28, 2020 at 17:57

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .