# Does the Fourier expansion of the j-function have any prime coefficients?

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.

• 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. – Jeremy Rouse Nov 21 '20 at 16:17
• 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 – Gerry Myerson Nov 21 '20 at 23:20
• @JeremyRouse In fact, given that growth rate, our naive expectation should be infinitely many prime coefficients. – JoshuaZ Nov 25 '20 at 21:25
• I discovered a bug in my program (a type error) that leads to all the coefficients being labelled "not prime". – Jeremy Rouse Nov 25 '20 at 22:21
• 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. – Sylvain JULIEN Nov 25 '20 at 22:37

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

• FWIW: Mathematica confirms that the two numbers listed above are prime. – Aeryk Nov 25 '20 at 19:31
• @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. – Wojowu Nov 25 '20 at 19:59
• 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). – Fredrik Johansson Nov 25 '20 at 20:04
• Fantastic. Thanks. Proving now that infinitely many such primes exist seems very difficult. – David Feldman Nov 25 '20 at 20:45
• 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). – Jeremy Rouse Nov 28 '20 at 17:57