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.

$\endgroup$
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. $\endgroup$ 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 $\endgroup$ 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
    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". $\endgroup$ 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. $\endgroup$ Nov 25, 2020 at 22:37

1 Answer 1

22
$\begingroup$

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

$\endgroup$
13
  • 3
    $\begingroup$ FWIW: Mathematica confirms that the two numbers listed above are prime. $\endgroup$
    – Aeryk
    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
    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). $\endgroup$ Nov 25, 2020 at 20:04
  • 4
    $\begingroup$ Fantastic. Thanks. Proving now that infinitely many such primes exist seems very difficult. $\endgroup$ 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). $\endgroup$ Nov 28, 2020 at 17:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.