urbit/base/pub/doc/hoon/library/3ba.md

section 3bA, lite number theory

++fu

++  fu                                                  ::  modulo (mul p q)
  |=  a=[p=@ q=@]
  =+  b=?:(=([0 0] a) 0 (~(inv fo p.a) (~(sit fo p.a) q.a)))
  |%

XX document

++dif

  ++  dif
    |=  [c=[@ @] d=[@ @]]
    [(~(dif fo p.a) -.c -.d) (~(dif fo q.a) +.c +.d)]
  ::

XX document

++exp

  ++  exp
    |=  [c=@ d=[@ @]]
    :-  (~(exp fo p.a) (mod c (dec p.a)) -.d)
    (~(exp fo q.a) (mod c (dec q.a)) +.d)
  ::

XX document

++out

  ++  out                                               ::  garner's formula
    |=  c=[@ @]
    %+  add
      +.c
    (mul q.a (~(pro fo p.a) b (~(dif fo p.a) -.c (~(sit fo p.a) +.c))))
  ::

XX document

++pro

  ++  pro
    |=  [c=[@ @] d=[@ @]]
    [(~(pro fo p.a) -.c -.d) (~(pro fo q.a) +.c +.d)]
  ::

XX document

++sum

  ++  sum
    |=  [c=[@ @] d=[@ @]]
    [(~(sum fo p.a) -.c -.d) (~(sum fo q.a) +.c +.d)]
  ::

XX document

++sit

  ++  sit
    |=  c=@
    [(mod c p.a) (mod c q.a)]

XX document