Monday, May 29, 2017

How to use BigZ - part 3

The binomial coefficients for big integers

The number of possibilities to choose k objects from n objects soon get to big for a single cell number. The word bschoose gives a big integer result for single cell inputs.

: bschoose \ n k -- b

  bone 0
  ?do dup i - bs*
     i 1+ bs/mod drop
  loop drop ;


2000 500 bschoose cr b.
5648284895675941420424412140748481039502890353942825357221051675360331984776743417002364625179991976070068866284527555107208940603781511988000970130381311935878493235111594076219803768997324618773852975824828528735285833615310777764160933348372329757027402537319600321600269195597902747298520883357267710485334098751949232380773741897267988881873218260056305793069941805234442045890109611836653468404129012879905442075185208447514284775689056520318572740750419026192611832748925888424320  ok

This word produce big integers with single cell factors that can be analysed by the word 

sfacset \ b -- b' set

2000 1000 bschoose sfacset bdrop cr zet.

{2,5,7,11,13,17,19,23,37,41,43,53,59,67,73,79,101,103,113,127,131,149,151,167,173,179,181,211,251,257,263,269,271,277,281,283,337,347,349,353,359,367,373,379,383,389,397,503,509,521,523,541,547,557,563,569,571,577,587,593,599,601,607,613,617,619,631,641,643,647,653,659,661,1009,1013,1019,1021,1031,1033,1039,1049,1051,1061,1063,1069,1087,1091,1093,1097,1103,1109,1117,1123,1129,1151,1153,1163,1171,1181,1187,1193,1201,1213,1217,1223,1229,1231,1237,1249,1259,1277,1279,1283,1289,1291,1297,1301,1303,1307,1319,1321,1327,1361,1367,1373,1381,1399,1409,1423,1427,1429,1433,1439,1447,1451,1453,1459,1471,1481,1483,1487,1489,1493,1499,1511,1523,1531,1543,1549,1553,1559,1567,1571,1579,1583,1597,1601,1607,1609,1613,1619,1621,1627,1637,1657,1663,1667,1669,1693,1697,1699,1709,1721,1723,1733,1741,1747,1753,1759,1777,1783,1787,1789,1801,1811,1823,1831,1847,1861,1867,1871,1873,1877,1879,1889,1901,1907,1913,1931,1933,1949,1951,1973,1979,1987,1993,1997,1999} ok

The word bsetprod calculates the big product of the singles in set

: bsetprod \ set -- b
  bone                   \ big one
  foreach                \ make ready for do-loop
  ?do zst> bs* loop ;

and can be used to calculate the radical for big integers with single cell factors:

: bsradical \ b -- b'

  sfacset bdrop 
  bsetprod ;

50 25 bschoose bsradical cr b.

1504888171878  ok

ErdÅ‘s squarefree conjecture (proved 1996) states that the central binomial coefficient (2n)Cn is not squarefree if n>4. The word sqrfacset calculates the set of all factors that occurs more than once: 

: sqrfacset \ b -- set

  bdup bsradical b/
  sfacset bdrop ;

20000 10000 bschoose sqrfacset cr zet.

{2,3,7,11,23,29,41,47,53,61,71,73,79,109,127,137,139} ok

The word

: maxel \ set -- n   non e
  zst> zst@ swap >zst zdrop ;

gives the maximal element in a set of integers.

: erdprime \ n -- p
  dup 2* swap bschoose
  sqrfacset maxel ;

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