Heights of cyclotomic polynomials  of large order.

Michael Monagan

Department of Mathematics,

Simon Fraser University.

The work is supported by the MITACS NCE of Canada and NSERC of Canada.

Our project is entitled "Mathematics of Computer Algebra and Analysis (MOCAA)".

Let  denote the cyclotomic polynomial of order k.  This is a monic polynomial with integer coefficients.  It is the irreducible factor of the polynomial  with complex roots the primitive k'th roots of unity.  For example, the first few cyclotomic polynomials are

 > for k to 6 do     Phi[k](x) = numtheory[cyclotomic](k,x); od;

Let  denote the height of the kth cyclotomic polynomial, that is, the magnitude of the largest coefficient of .  It is well known that  for all  and that for  we have

 > Phi105 := numtheory[cyclotomic](105,x);

 > H105 := maxnorm(Phi105);

Below is a table of increasing heights  for  .

 > array([[k,Hk],  [105=ifactor(105),2],  [385=ifactor(385),3],  [1365=ifactor(1365),4],  [1785=ifactor(1785),5],  [2805=ifactor(2805),6],  [3135=ifactor(3135),7],  [6545=ifactor(6545),9],  [10465=ifactor(10465), 4],  [11305=ifactor(11305), 23],  [17255=ifactor(17255), 25],  [20615=ifactor(20615), 27],  [26565=ifactor(26565), 59],  [40755=ifactor(40755), 359], [106743, 397], [171717, 434], [255255, 532], [279565, 585], [285285, 1182], [327845, 31010], [707455, 35111], [886445, 44125], [983535, 59518], [1181895, 14102773], [1752465, 14703509], [3949491, 56938657], [8070699, 74989473], [10163195, 1376877780831], [13441645, 1475674234751], [15069565, 1666495909761], [30489585, 2201904353336], [37495115, 2286541988726], [40324935, 2699208408726], [43730115, 862550638890874931], [169828113, 31484567640915734951] ]);

 > matrix([[7, Hk], [105 = ``(3)*``(5)*``(7), 2], [385 = ``(5)*``(7)*``(11), 3], [1365 = ``(3)*``(5)*``(7)*``(13), 4], [1785 = ``(3)*``(5)*``(7)*``(17), 5], [2805 = ``(3)*``(5)*``(11)*``(17), 6], [3135 = ``(3)*``(5)*``(11)*``(19), 7], [6545 = ``(5)*``(7)*``(11)*``(17), 9], [10465 = ``(5)*``(7)*``(13)*``(23), 4], [11305 = ``(5)*``(7)*``(17)*``(19), 23], [17255 = ``(5)*``(7)*``(17)*``(29), 25], [20615 = ``(5)*``(7)*``(19)*``(31), 27], [26565 = ``(3)*``(5)*``(7)*``(11)*``(23), 59], [40755 = ``(3)*``(5)*``(11)*``(13)*``(19), 359], [106743, 397], [171717, 434], [255255, 532], [279565, 585], [285285, 1182], [327845, 31010], [707455, 35111], [886445, 44125], [983535, 59518], [1181895, 14102773], [1752465, 14703509], [3949491, 56938657], [8070699, 74989473], [10163195, 1376877780831], [13441645, 1475674234751], [15069565, 1666495909761], [30489585, 2201904353336], [37495115, 2286541988726], [40324935, 2699208408726], [43730115, 862550638890874931], [169828113, 31484567640915734951]]);

 > matrix([[7, Hk], [105 = ``(3)*``(5)*``(7), 2], [385 = ``(5)*``(7)*``(11), 3], [1365 = ``(3)*``(5)*``(7)*``(13), 4], [1785 = ``(3)*``(5)*``(7)*``(17), 5], [2805 = ``(3)*``(5)*``(11)*``(17), 6], [3135 = ``(3)*``(5)*``(11)*``(19), 7], [6545 = ``(5)*``(7)*``(11)*``(17), 9], [10465 = ``(5)*``(7)*``(13)*``(23), 4], [11305 = ``(5)*``(7)*``(17)*``(19), 23], [17255 = ``(5)*``(7)*``(17)*``(29), 25], [20615 = ``(5)*``(7)*``(19)*``(31), 27], [26565 = ``(3)*``(5)*``(7)*``(11)*``(23), 59], [40755 = ``(3)*``(5)*``(11)*``(13)*``(19), 359], [106743, 397], [171717, 434], [255255, 532], [279565, 585], [285285, 1182], [327845, 31010], [707455, 35111], [886445, 44125], [983535, 59518], [1181895, 14102773], [1752465, 14703509], [3949491, 56938657], [8070699, 74989473], [10163195, 1376877780831], [13441645, 1475674234751], [15069565, 1666495909761], [30489585, 2201904353336], [37495115, 2286541988726], [40324935, 2699208408726], [43730115, 862550638890874931], [169828113, 31484567640915734951]]);

The value  for   is the first k for which  .

The value
for k=43730115 is the first k for which  .

The value  =  31484567640915734951 for   was computed by Andrew Arnold in 2008  .

The value  for   was computed by Koshiba Yoichi.

The value    for     was computed by Michael Monagan in 2007.

The value     for     was computed by Michael Monagan in 2007.  This is for the product of the first 9 primes.

The largest  computed so far by Michael Monagan is for  = .  The value of  is 64540997036010911566826446181523888971563 which is a 135 bit integer which is > .  This was found by adding additional primes to the case 43730115.

Some other values of high order that we've computed are

 > Digits := 4: Matrix([['k','Hk','log[2]'(Hk)],  [3234846615, 2888582082500892851, log[2.0](2888582082500892851)],  [43730115, 862550638890874931, log[2.0](862550638890874931)],  [1005792645, 1265789099436496061273, log[2.0](1265789099436496061273)],  [1880394945, 64540997036010911566826446181523888971563,       log[2.0](64540997036010911566826446181523888971563)] ]);

 >