Western Number Theory Problems, 17 & 19 Dec 2016

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1 Wester Number Theory Problems, 7 & 9 Dec 6 for distributio prior to 7 (Pacific Grove) meetig Edited by Gerry Myerso based o otes by Kjell Woodig Summary of earlier meetigs & problem sets with old (pre 984) & ew umberig. 967 Berkeley 968 Berkeley 969 Asilomar 97 Tucso 97 Asilomar 97 Claremot 7: 7:5 97 Los Ageles 7: 7:6 974 Los Ageles 74: 74:8 975 Asilomar 75: 75: 976 Sa Diego 65 i.e., 76: 76: Los Ageles 48 i.e., 77: 77: Sata Barbara 5 87 i.e., 78: 78:7 979 Asilomar i.e., 79: 79: 98 Tucso 5 68 i.e., 8: 8:8 98 Sata Barbara 8 i.e., 8: 8:8 98 Sa Diego 5 75 i.e., 8: 8:5 98 Asilomar 4 48 i.e., 8: 8:8 984 Asilomar 84: 84:7 985 Asilomar 85: 85: 986 Tucso 86: 86: 987 Asilomar 87: 87:5 988 Las Vegas 88: 88: 989 Asilomar 89: 89: 99 Asilomar 9: 9:9 99 Asilomar 9: 9:5 99 Corvallis 9: 9:9 99 Asilomar 9: 9: 994 Sa Diego 94: 94:7 995 Asilomar 95: 95:9 996 Las Vegas 96: 96:8 997 Asilomar 97: 97: 998 Sa Fracisco 98: 98:4 999 Asilomar 99: 99: Sa Diego : :5 Asilomar : : Sa Fracisco : :4 Asilomar : :8 4 Las Vegas 4: 4:7 5 Asilomar 5: 5: 6 Eseada 6: 6:5 7 Asilomar 7: 7:5 8 Fort Collis 8: 8:5 9 Asilomar 9: 9: Orem : : Asilomar..6 Asilomar : :7 Asilomar.. 4 Pacific Grove 4: 4: 5 Pacific Grove 5: 5:5 6 Pacific Grove 6: 6:4 COMMENTS ON ANY PROBLEM WELCOME AT ANY TIME Departmet of Mathematics, Macquarie Uiversity, NSW 9 Australia gerry.myerso@mq.edu.au Australia fa

2 Problems proposed 7 ad 9 December 6 6: (Alessadro Rezede de Macedo) Let k() = P d dk. How does k fail to be ijective, as a fuctio of k? Remarks:. Carl Pomerace oted that Erdős, O the ormal umber of prime factors of p ad some related problems cocerig Euler s '-fuctio, Quart. J. Math., Oford Ser. 6 (95) 5, Zetralblatt, 49, available as 95-8 at p erdos/erdos.html, used shifted primes to show that () is far from ijective. Carl otes that the same proof, with chaged to +, works for, ad suggested this would be hard to do for k> you would eed to show somethig like p + is smooth for a positive proportio of primes p. See also C. Pomerace, Two methods i elemetary aalytic umber theory, i R. A. Molli, ed., Number Theory ad Applicatios, Kluwer Academic Publishers, Dordrecht 989, pp We ote that (6) = (7) (ad (6) = (7) for all with gcd(, 4) = ); (4) = (6) (ad (4) = (6) for gcd(, 78) = ); (4) = (47) (ad (4) = (47) for gcd(, 47) = ); () = () = (4); (6) = (57); (86) = (5) = (7); () = (49); ad so o. Pairs m, with (m) = () are tabulated at due to Ma Alekseyev. The smallest etry is (8496) = (945) = It follows that the set of m for which there eists >mwith (m) = () has positive lower desity. 6: (David Bailey) Let w (t) = X m,...,m m t mt mt (m + + m ) t cotiued aalytically to a meromorphic fuctio o the plae.. Are there ratioals r,...,r k such that w () = ± log + P k r j ( j)?. If so, what are the ratioals? Remark: This has bee settled for the first two cases, ad there is umerical evidece for may more. 6: (Ma Alekseyev, via Gerry Myerso) Let =, ad for k let k+ = k +5. The we have k k+ for k =,,, 4. Does this divisibility hold for all k? The questio appears, with some discussio, at ad also at ad ad (i Russia) at Remark: Carl Pomerace suggests lookig at such that + 5 to see what patters there may be. (cotributed by Ma Alekseyev) tabulates such that +. From this, the such that + 5 start with,, 9, 6, 6, 577, 7, , , ,

3 6:4 (Carl Pomerace) Let S = : divides This set is ifiite, but misses a positive proportio of positive itegers. I particular, it misses all those with a prime factor eceedig p, a set of desity log. O the other had, if = pq where p ad q are prime ad.5p <q<p, the is i S. DoesS cotai a positive proportio of the positive itegers? Does S have a desity? Remarks:. These umbers are tabulated at Pate Staica cojectures that for 7 we have apple #S apple (log log ) (log log ) 6:5 (Adrew Shallue) Pomerace, O the distributio of pseudoprimes, Math. Comp. 7 (98) (see display (5), see also referece to Pomerace paper at 6:), proved that, for fied, ad for su cietly large, #{ m apple : (m) = }applel() +o() Here (m) =lcm p m (p ) is Carmichael s fuctio, ad log log log log L() =ep log log Also, where (m) =lcm p m (p ). #{ m apple : (m) = }applel() +o() Ca oe improve this, perhaps o average? 6:6 (Neville Robbis) A osolo partitio of is a partitio i which each part occurs at least twice. The umber of osolo partitios of equals the umber of partitios of ito parts ot ± mod 6. For d, fid a similar equality for the umber of partitios of i which each part occurs at least d times. Remarks:. The umber of osolo partitios is tabulated at The umber of osolo partitios is also the umber of partitios ito parts, each larger tha oe, ad di erig by at least two.. Simo Rubistei-Salzedo suggests cosultig Bruce Berdt s book, Number Theory i the Spirit of Ramauja. 6:7 (Sugji Kim) Are the matrices, A, all osigular? The etries here are the first 4, 9, 6,... primes, etered i order, left to right ad the top to bottom.

4 Remarks:. This is essetially proposed by user Widawese, but there the primes are take to start from.. Nosigularity has bee verified for both versios of the questio up to size by user Peter.. Reate Scheidler asked whether there is aythig special here about primes, ad whether oe could just choose radomly icreasig itegers, otig that a radom iteger matri will be osigular. [I suppose it depeds o what s meat by radom, e.g., is 5 7 A Simo Rubistei-Salzedo asks whether the determiats are always egative, ad whether they are icreasig i absolute value. 6:8 (Simo Rubistei-Salzedo) A umber is practical if each umber less tha is a sum of distict divisors of. For k 4, do there eist ifiitely may such that, +,...,+ (k ) are all practical? Remark: This is true for k = (G. Melfi, O two cojectures about practical umbers, J. Number Theory 56 (996) 5, MR 7 (96i:6)). 6:9 (Bart Goddard) Is there ay arragemet of the first primes i a matri such that the matri is sigular? (Compare 6:7) Solutio: Kevi McGow foud det A = for A 5 7 A 7 9 6: (Christia Ballot) Let =(+ p 5)/. Cosider a tree built as follows. The first level cosists of the umber. Each umber r at each level has the two descedats r + ad r, ecept that we prue ay umber that has already appeared. Thus, the secod level has just the umber ; the third level has ad ; the fourth level has,, ad + (sice = + ). The umber of umbers o each level of the tree, G, forms a sequece startig,,,, 5, 8, but G 7 =, so it s ot the Fiboaccis. It is cojectured that G + = G + + G for. Remark: The sequece is tabulated at 6: (Carter Smith) Defie S : N! R by S() =( p, ) i polar coordiates, S() =( p cos, p si ) i rectagular, where is the golde ratio. The the poits correspodig to prime values of lie o specific rays. As varies, the umber of rays varies, ad the picture chages, drastically. Remarks:. Adrew Shallue refereces Matt Boelkis pleary talk, Fiboacci s Garde, at the meetig of the Illiois Sectio of the MAA i 6.. Sugji Kim suggests the pictures for ratioal ad irratioal will be very di eret. 4

5 6: (Paul Youg) For i C, defie D () It is kow that if > is odd, ad d ( < < thed () 6= C.... A ), the D (d) =. Show that for comple if Remark: This is kow to be true if <( + log ), ad has bee verified computatioally for all <. a b 6: (Elie Alhajjar) Let A = with iteger etries, bc 6=, = ad bc. Let c d A =. Is there a formula for gcd(, ) i terms of a, b, c, d,? Note that gcd(, ) divides. 6:4 (Berardo Recamá Satos, via Gerry Myerso) Is it possible to completely tile a square with di eret rectagles of iteger sides but all with the same area? This is Remarks:. rectagles. Without the iteger restrictio, there are may solutios. Oe has seve rectagles, each of area /7, with sides i Q( p 9).. Ed Pegg, Jr. suggests a umber of cadidate squares, e.g., a square of side 96 has area 475, ad there are 4 iteger rectagles of area