## Wednesday, June 30, 2021

### Polynomials in Z[X]

The separate stacks xst, yst and zst used for the implementation of nested sets, can also be used to dynamical allocation of arrays, here arrays for the coefficients of polynomials. Defining a non nested list of integers

( 3 0 -2 7 -3 1 )  ok

works as long as the list not is interpreted as nested, when the negative numbers signalize a count. (All number elements of nested sets or lists must be non negative.)

Interpreted as non nested linear lists all set stack manipulation words works, so the administration of these polynomials works as  usual in Forth. The list above correspond to the polynomial

( 3 0 -2 7 -3 1 ) poly. 3-2x²+7x³-3x⁴+x⁵ ok

\ Polynomials

: >da \ vect -- vect ad n
zst @ zst@ cs tuck cells - swap ;
\ Gives the address to the first coefficient plus the count
\ of the polynomial at top of stack

: >da2 \ vect2 vect1 -- vect2 vect1 ad2 n2
\ Gives address and count to the second polynomial of stack

: >zst+ \ vect1 m -- vect2
zst> swap >zst 2 - >zst ;
\ Add item to the list

: da \ -- vect ad
-1 >zst zst @ ;
\ Initiate an empty list

: da. \ vect --
>da 0
do dup i cells + @ .
loop zdrop drop ;
\ print the coefficients

The word Z. cannot be used since it interpret the list as nested.

( 3 0 -2 7 -3 1 ) da. 3 0 -2 7 -3 1  ok

Some systems can write exponents:

\ Printing polynomials

create potence
s" "   s, s" x"  s, s" x²" s, s" x³" s, s" x⁴" s,
s" x⁵" s, s" x⁶" s, s" x⁷" s, s" x⁸" s, s" x⁹" s,
s" x¹⁰" s, s" x¹¹" s, s" x¹²" s, s" x¹³" s, s" x¹⁴" s,

true value lowterm
: .term \ i n --
?dup 0= if drop exit then
dup 0<
if ." -"
else lowterm 0= if ." +" then
then abs dup 1 > 2 pick 0= or
if 0 <# #s #> type
else drop
then false to lowterm
cells potence + count type ;

: poly. \ vect --
true to lowterm
>da 0
do i over i cells + @ .term
loop zdrop drop ;

Since BigZ is limited to non negative integers greatest common divisor is defined for unsigned integers UGCD and a word GCD for all integers have to be defined:

: gcd \ n1 n2 -- n      \ Greatest common divisor
2dup or 0=
if 2drop 1 exit then
abs swap abs
2dup u< if swap then  \ smallest of a b on top of stack
?dup 0= if exit then  \ return second value if tos is zero
begin tuck            \ y x y first time b a b
0 swap um/mod      \ y x 0 y --> y r q
drop dup 0=        \ y r [r=0]
until drop ;          \ y

: multgcd \ k1...kn n -- gcd
1 do gcd loop ;
\ Gives multiple greatest common device

\ Calculation with polynomials

: polynom \ ad n m -- m'
locals| m | cells over + cell-
dup @ -rot cell-
?do m * i @ + -cell +loop ;
\ m' is the evaluation of m with polynomial at ad n

: polyn \ vect m -- vect m'
>da rot polynom ;
\ m' is the evaluation of m with the polynomial vect

: gcoeff \ vect -- vect n
zst @ cell - @ ;
\ Gives the coefficient of the greatest power

: rrzs \ vect1 -- vect2  "reduce right zeroes"
begin gcoeff 0=
while zst> zst> drop 2 + >zst
repeat ;
\ Eliminate leading coefficient equal to zero

n1 n2 + 1- 0
do 0 i 1+ 0
do j i - 0 n2 within i n1 < and
if i cells ad1 + @
j i - cells ad2 + @ * +
then
loop >zst+
loop rrzs ;
\ Multiply polynomials given by arrays

: p* \ vect1 vect2 -- vect3
>da2 >da poly*
znip znip ;
\ Multiply polynomials

( 0 -1 2 1 ) zdup poly. -x+2x²+x³ ok
( 2 0 2 3 ) zdup poly. 2+2x²+3x³ ok
p* poly. -2x+4x²+x⁴+8x⁵+3x⁶ ok

: p+ \ vect1 vect2 -- vect3
zst>> cs 0
do ad i cells + +!
loop rrzs ;

: pnegate \ vect1 -- vect2
do dup i + dup @ negate swap ! cell
+loop drop ;
\ Negate a polynomial

: p- \ vect1 vect2 -- vect3
pnegate p+ ;
\ Subtract polynomials

: ps/ \ vect1 n -- vect2
locals| n |
>da cells over + swap
do i @ n / i ! cell +loop ;
\ Divide polynomial with integer

: makepoly \ vect ad n --  name of polynomial
cr ." : " type space
zst> zst> .
cs 1- 1
do ." over * " zst> . ." + "
loop ." * " zst> . ." + ; " ;
\ Prints the definition of a polynomial to be pasted

( 3 0 -2 7 -3 1 ) s" poly1" makepoly
: poly1 1 over * -3 + over * 7 + over * -2 + over * 0 + * 3 + ;  ok

Copying and pasting the output easy define the polynomial POLY1, eventually after some cleaning.

There is a nice theory of integer-valued polynomials by Pólya. That is, polynomial with rational coefficients that gives integer outputs for integer inputs. The set of these polynomials is a subring of Q[X], the ring of all rational polynomials. This subring is denoted int(Z).

This can be used to calculate the fixed prime divisors of polynomials with integer coefficients. Even if the coefficients of a polynomial in Z[X] has the greatest common divisor 1, all the outputs of the polynomial might be divisible by a fixed number. For example the polynomial

6-9x-2x²+5x⁴

always give an output that is divisible by 6, and therefore have the fixed prime divisors 2 and 3.

\ Integer valued polynomials

: bin*sum \ ad k -- sum
k 0= if 1 exit then 0 k 0
?do i cells ad + @
k i choose * +
loop ;
\ Calculate the sum in the figure above

: polyacoeff \ ad1 n1 -- vect
n1 1
loop ;
\ Calculate the vector (c0,...,cn) from
\ integer polynomial at ad1 n1

: polya \ ad n m -- m'
swap -rot locals| m ad | 0 swap 0
?do i cells ad + @
m i choose * +
loop ;
\ m' is the evaluation of m with the pólya function at ad n

: coeffgcd \ vect -- n
zst>> cs       \ CS transform set count into stack count
multgcd ;

: fixdiv \ vect -- vect n
>da            \ get address and count for polynomial
polyacoeff     \ calculate Pólya's coefficients
coeffgcd ;
\ The multiple GCD of c0,...,cn is the fixed divisor of the
\ corresponding original polynomial with integer coefficients

Eisensteins criteria: If there exist a prime number p which not divides an but a0,...,an-1, and p² not divide a0, then
a0+a1*x+a2*x²+...+an*x^n is irreducible over the rational numbers.

\ Eisenstein's criteria

: iseisenstein \ vect -- vect flag   "is an Eisenstein polynomial?"
zdup zst> 2 + zst> abs false 0 locals| p flag  an |
>zst zst>> cs multgcd abs primes ?dup
if 0
do to p flag 0=
if an p umod 0= 0=
>da drop @ abs p ^2 umod 0= 0= and
to flag
then
loop
then flag ;

Most polynomial are irreducible but very few are Eisenstein.

Theorem 1 (Chen & al): If the polynomial f(X) in Z[X] is reducible then the number of positive primes of the form f(a) is less then or equal the degree of f(X). For all degrees n there is a reducible polynomial f(X) with different a1,...,an such that f(ai) is a positive prime.

So finding a polynomial giving primes for n+1 different values is finding an irreducible polynomial. But a lot of polynomials have a fixed divisor greater than one, and those can't be proved irreducible by the theorem above. A more relevant test is therefore to combine theorem 1 with the Pólya fix divisor test.

: ischen \ vect -- vect flag  "false may be and true is irreducible"
fixdiv 1 > if false exit then
degree locals| n |
0 bits n / 2e s>f f** f>s 1000 min 0
do i polyn dup 0>
if isprime -
dup n > if leave then
else drop
then
i negate polyn dup 0>
if isprime -
dup n > if leave then
else drop
then
loop n > ;

\ Polynomials

\ Dynamical allocation of arrays

: >da \ vect -- vect ad n
zst @ zst@ cs tuck cells - swap ;
\ Gives the address to the first coefficient plus the count
\ of the polynomial at top of stack

: >da2 \ vect2 vect1 -- vect2 vect1 ad2 n2
\ Gives address and count to the second polynomial of stack

: >xst+ \ vect1 m -- vect2
xst> swap >xst 2 - >xst ;
\ Add item to the xst list

: >zst+ \ vect1 m -- vect2
zst> swap >zst 2 - >zst ;
\ Add item to the zst list

: da \ -- vect ad
-1 >zst zst @ ;
\ Initiate an empty list

: da. \ vect --
>da 0
do dup i cells + @ .
loop zdrop drop ;
\ print the coefficients

\ Printing polynomials
\ 64 bits systems only

create potence
s" " s,    s" x" s,   s" x²" s,  s" x³" s,  s" x⁴" s,
s" x⁵" s,  s" x⁶" s,  s" x⁷" s,  s" x⁸" s,  s" x⁹" s,
s" x¹⁰" s, s" x¹¹" s, s" x¹²" s, s" x¹³" s, s" x¹⁴" s,
s" x¹⁵" s, s" x¹⁶" s, s" x¹⁷" s, s" x¹⁸" s, s" x¹⁹" s,

true value lowterm
: .term \ i n --
?dup 0= if drop exit then
dup 0<
if ." -"
else lowterm 0= if ." +" then
then abs dup 1 > 2 pick 0= or
if 0 <# #s #> type
else drop
then false to lowterm
cells potence + count type ;

: p. \ vect --
true to lowterm
>da 0
do i over i cells + @ .term
loop zdrop drop ;

\ Greatest common divisors for multiple integers

: multgcd \ k1...kn n -- gcd
dup 0= if exit then
swap abs swap 1
?do swap abs ugcd loop ;
\ Gives multiple greatest common device

\ Calculation with polynomials

: polynom \ ad n m -- m'
locals| m | cells over + cell-
dup @ -rot cell-
?do m * i @ + -cell +loop ;
\ m' is the evaluation of m with polynomial at ad n

: sbs* \ sb m -- sb*m
dup 0< xs> xor >xs abs bs* ;

: s>sb \ n -- sb
dup abs s>b 0< >xs ;

: sbpolynom \ ad n m -- sb
locals| m | cells over + cell-
dup @ s>sb cell-
?do m sbs* i @ s>sb sb+ -cell +loop ;
\ single input and big output

: polyn \ vect m -- vect m'
>da rot polynom ;
\ m' is the evaluation of m with the polynomial vect

: sbpolyn \ vect m -- vect sb
>da rot sbpolynom ;
\ m' is the evaluation of m with the polynomial vect

: gcoeff \ vect -- vect n
zst @ cell - @ ;
\ Gives the coefficient of the greatest power

: lcoeff \ vect -- vect n
>da drop @ ;
\ Gives the coefficient of the constant term

: rrzs \ vect1 -- vect2  "reduce right zeroes"
begin gcoeff 0=
while zst> zst> drop 2 + >zst
repeat ;
\ Eliminate leading coefficient equal to zero

n1 n2 + 1- 0
do 0 i 1+ 0
do j i - 0 n2 within i n1 < and
if i cells ad1 + @
j i - cells ad2 + @ * +
then
loop >zst+
loop rrzs ;
\ Multiply polynomials given by arrays

: p* \ vect1 vect2 -- vect3
>da2 >da poly*
znip znip ;
\ Multiply polynomials

: v+ \ vect1 vect2 -- vect3
zst>> cs 0
do ad i cells + +!
loop ;

: p+ \ vect1 vect2 -- vect3
v+ rrzs ;

: ps* \ vect1 n -- vect2
locals| n |
>da cells over + swap
do i @ n * i ! cell +loop ;
\ Multiply polynomial with integer

: pnegate \ vect1 -- vect2
-1 ps* ;

false [if]
: pnegate \ vect1 -- vect2
do dup i + dup @ negate swap ! cell
+loop drop ;
\ Negate a polynomial
[then]

: p- \ vect1 vect2 -- vect3
pnegate p+ ;
\ Subtract polynomials

: v- \ vect1 vect2 -- vect3
pnegate v+ ;
\ Subtract vectors

: ps/ \ vect1 n -- vect2
locals| n |
>da cells over + swap
do i @ n / i ! cell +loop ;
\ Divide polynomial with integer

: degree \ vect -- vect n
zst@ cs 1- ;

\ long division

: vcutr \ vect1 n -- vect2
degree swap 1- -
>r zst>> cs r@ - >xs
r> drops xs> 2* 1+ negate >>zst ;
\ vect2 is the n rightmost coefficients of vect1

: vshiftr \ vect1 -- vect2
zst> zst> drop 2 + >zst ;
\ drop the rightmost coefficient

: getcoeff \ xvect i -- xvect n
cells xst @ cell - swap - @ ;

: vor \ vect -- flag
zst>> cs 1 ?do or loop ;

: ldivide \ -- q r
zst> zst@ swap >zst
yst> yst@ swap >yst
/mod swap ;

: lclean \ --
xst setdrop yst setdrop ;

: lbuild \ v n q -- v' n+1
dup >xs 1 under+
yst zst setcopy ps* v- ;

: lnodiv \ --
drop 0
?do xsdrop loop false ;

: p/ \ v1 v2 -- v1/v2 flag
false locals| flag |
degree zst yst setmove
degree zst xst setcopy
over 1+ vcutr                           \ w
2 + 2 under+ swap 0 -rot
do ldivide
if true to flag leave then
lbuild
vshiftr ( i getcoeff ) zswap vmerge  \ w'
loop flag if lclean lnodiv exit then
ldivide if lclean lnodiv exit then
lbuild vor lclean
( over 0 ?do xs> loop ) nip 0= ;
\ flag is true if v2 divides v1
\ else result is irrelevant

\ auto definition of polynomial
: makepoly \ vect ad n --  name of polynomial
cr ." : " type space
zst> zst> .
cs 1- 1
do ." over * " zst> . ." + "
loop ." * " zst> . ." + ; " ;
\ Prints the definition of a polynomial to be
\ copied and pasted

\ Integer valued polynomials

: bin*sum \ ad k -- sum
k 0= if 1 exit then 0 k 0
?do i cells ad + @
k i choose * +
loop ;

: polyacoeff \ ad1 n1 -- vect
n1 1
loop ;
\ Calculate the vector (c0,...,cn) from
\ integer polynomial at ad1 n1

: polya \ ad n m -- m'
swap -rot locals| m ad | 0 swap 0
?do i cells ad + @
m i choose * +
loop ;
\ m' is the evaluation of m with the pólya function at ad n

: coeffgcd \ vect -- n
zst>> cs       \ CS transform set count into stack count
multgcd ;
\ GCD of the coefficients

: fixdiv \ vect -- vect n
>da            \ get address and count for polynomial
polyacoeff     \ calculate Pólya's coefficients
coeffgcd ;
\ The multiple GCD of c0,...,cn is the fixed divisor of the
\ corresponding original polynomial with integer coefficients

: divcofac \ vect -- vect'
zdup coeffgcd ps/ ;

: iseisenstein \ vect -- vect flag
zdup zst> 2 + zst> abs false 0 locals| p flag  an |
>zst coeffgcd dup an ugcd 1 <>
if zdrop drop false exit then
primes ?dup
if 0
do to p flag 0=
if an p umod 0= 0=
>da drop @ abs p ^2 umod 0= 0= and
to flag
then
loop
then flag ;

2000 value xlim

: isirr \ vect -- vect flag
iseisenstein if true exit then
degree 0= if gcoeff isp exit then
degree 1 = if zdup coeffgcd 1 = exit then
fixdiv degree 0 0 locals| posp negp n d |
0 sbpolyn d bs/mod drop bisprime
if xs@ if negp 1+ to negp else posp 1+ to posp then
then xsdrop
xlim 1
do i sbpolyn d bs/mod drop bisprime
if xs@ if negp 1+ to negp else posp 1+ to posp then
then xsdrop
i negate sbpolyn d bs/mod drop bisprime
if xs@ if negp 1+ to negp else posp 1+ to posp then
then xsdrop
posp n > negp n > or if leave then
loop posp n > negp n > or ;

: nopsqr \ x p -- x'     p|x
begin 2dup /mod swap 0=
while -rot nip
repeat drop * ;

: negate? \ |n| -- n
2 random if negate then ;

: pickprime \ n -- p
primes dup >r 1 max random
pick r> drops ;

: geteis0 \ u -- vect p
( )
2 - 1 max random 2 +
dup pickprime
tuck nopsqr negate? >zst+ ;

: x/p^n \ an p -- an'
begin 2dup mod 0=
while tuck / swap
repeat drop ;

: geteisvar \ n u -- vect
dup geteis0 locals| p u | 1- 1 max random 1+ 0
?do u p / random 1+ p * negate? >zst+
loop u 1+ random 1+ p x/p^n dup 0= or
negate? >zst+
divcofac ;

: dupderiv \ vect -- vect vect'
( >da swap locals| ad | 1
do ad i cells + @ i * loop ) ;

: deriv \ vect -- vect'
dupderiv znip ;

\ p(x) --> p(x+d)

: mtransl \ k d ak -- vect
locals| ak d k |
( k 1+ 0
do k i choose d i ** * ak *
loop ) ;

: zerovect \ n -- vect
>r ( r> 0
do 0 loop ) ;

: ptransl \ vect1 d -- vect2
locals| d |
>da 0 over zerovect
do i over i cells + @ d swap
mtransl p+
loop drop znip ;

\ Rational roots

: q* \ a b c d -- ac/(ac,bd) bd/(ac,bd)
rot * >r * r>         \ ac bd
2dup abs swap abs
ugcd tuck             \ ac gcd bd gcd
/ >r / r> ;

: q/  2swap q* ;

: q+ \ a b c d -- (ad+bc)/gcd bd/gcd
dup 3 pick * >r      \ a b c d  r: bd
-rot * -rot * +      \ a*d+b*c  r: bd
dup abs r@ abs
ugcd r> over         \ a*d+b*c gcd bd gcd
/ >r / r> ;

: q-  negate q+ ;

: qpolynom \ ad n a b -- a' b'
locals| b a | cells over + cell-
dup @ >r cell- r> 1 2swap
do a b q* i @ 1 q+ -cell +loop ;

: getpospairs \ vect -- vect set
lcoeff abs gcoeff abs divz divz
cartprod ;

: getypair \ yset -- yset' y x
yst> drop yst> yst> ;

: haverationalroots \ vect -- vect flag
lcoeff 0= if true exit then
getpospairs zst yst setmove
begin yst@
while ysplit
getypair 2dup ugcd 1 =
if >da 2over qpolynom drop 0=
if yst setdrop 2drop true exit then
>r negate >r
>da r> r> qpolynom drop 0=
if yst setdrop true exit then
else 2drop
then
repeat yst> ;

: setofroots \ vect -- vect set
lcoeff 0= if true exit then
getpospairs
zst yst setmove xst @
begin yst@
while ysplit
getypair 2dup ugcd 1 =
if >da 2over qpolynom drop 0=
if ( 2dup ) zst xst setmove then
swap negate swap
>da 2over qpolynom drop 0=
if ( 2dup ) zst xst setmove then
then 2drop
repeat yst> drop
xst @ - cell / 2* >xst
xst zst setmove ;

: .root \ b a --  "a/b"
dup 0= if . drop exit then
over abs 1 = if . drop exit then
. 8 emit ." /" . ;

: .roots \ set --
zst> cs 3 / 0
do zst> drop zst> zst> .root space loop
;

: isirreducible \ vect -- vect flag
haverationalroots degree 1 > and
if false else isirr then ;

## Saturday, January 18, 2020

The Pollard rho algorithm factorize a composite number n in a time proportional to √p, where p is the smallest prime that divides n. In worst case p≈√n so the algorithm factorize proportional to ∜n. Compare with trial-and-error that factorize proportional to √n.

The algorithm is easy to implement and is built upon an the greatest common divisor gcd (which can be calculated by the algorithm of Euclide) and a polynomial P(x) of degree >1. Most common polynomial to use is P(x)=x²+1. Let Pn(x)=P(x) mod n, that is the rest when P(x) is divided by n. Pn is a function Pn:ℕ→ℤn, where ℤn={0,1,2,...,n-1}. The function Pn will act like a simple pseudo random generator in the algorithm.

Define Xi+1=Pn(Xi). Since ℤn is a finite set there are smallest i,j such that Xi=Xi+j, when the sequence start to repeat itself. Mostly the sequence Xi isn't cyclic from the start Xo, but later on at some Xi. There is at trick to find out if Xi is in the loop or not. It's like sending away a turtle and a hare on the same track at the same time. When the hare again is comming besides the turtle they must both be in the loop.

Therefore, define a second sequence Yi+1=Pn(Pn(Yi)), where Xo=Yo. When Yi=Xi, i>0, Xi and Yi are in the cycle. But for Pollard rho the real important sequences are the uncalculated sequences Vi+1=Pm(Vi) and Wi+1=Pm(Pm(Wi)) where Vo=Wo=Xo and m|n (which aren't calculated since we don't know any m|n yet). When Wi=Vi, i>0, then m|Xi-Yi and gcd(Xi-Yi,n)=m. If m=1 the result is neglected and the process go on with Xi+1 and Yi+1.
If m=n the start values Xo=Yo is said to fail and may be increamented by 1 for an other try. Now, gcd(Xi-Yi,n)=n when Xi and Yi become equal before any of the uncalculated sequences Vi and Wi become equal for some non trivial divisor m>1 of n. There could still be non trivial divisors to obtain for gcd(Xj-Yj,n) for j>i when Xi=Yi, but there is no guarranty and Xi=Yi is a natural terminal case.

An example with P(x)=x²+1 that is factorized while continuing after failure for Xo=2 is n=4294952621. An example that can't be factorized after failure with Xo=2 but with Xo=3 is n=4294939069.

Increasing Xo after termination when gcd(Xi-Yi,n)=n will always find a factor of n, because if m is a non trivial factor of n, then Xo=m immediately gives m as a factor. That is, for
P(x)=x²+1.

## Friday, September 1, 2017

### BigZ - new top level instructions

I've reorganized and extended some top level words:

create-set \ m n xt -- set
filter-set \ set1 xt -- set2
build-set \ m n xt -- set
transform-set \ set1 xt -- set2

This words should be used with the word :| to define and form sets.

1 10 :| b a | a ^2 b ^2 + isprime ; create-set zdup cr zet.
{(1,1),(2,1),(4,1),(6,1),(1,2),(3,2),(5,2),(7,2),(2,3),(8,3),(1,4),(5,4),(9,4),(2,5),(4,5),(6,5),(8,5),(1,6),(5,6),(2,7),(8,7),(3,8),(5,8),(7,8),(4,9)} ok

:| b a | a ^2 b ^2 + ; transform-set zdup cr zet. {2,5,13,17,29,37,41,53,61,73,89,97,113} ok

:| n | n 4 mod 3 = ; filter-set zet. 0 ok

The word :| define a nameless word and count the number of parameters.

: bl# \ ad n -- m    count the number of spaces in the string
over + swap 0 -rot
do i c@ bl = -
loop ;

variable loc#    \ the number of parameters
variable sta#    \ the number of outputs on the stack

: locals# \ --
>in @ >r
[char] | parse bl# loc# !
r> >in !
1 sta# ! ; immediate

: :| \ --
:noname
postpone locals#
postpone locals| ;

The nameless words, represented by xt on stack, could be of two types:

1. Taking parameters and leaving a flag.
2. Taking parameters and leaving a non negative integer.

In the first case a set with the "dimension" stated by the parameters is the result. This works with CREATE-SET and FILTER-SET.

In the second case a set of values is the result. This works with BUILD-SET and TRANSFORM-SET.

1 10 :| x | x ^2 ; build-set cr zet.
{1,4,9,16,25,36,49,64,81} ok

Normally these nameless words leave one integer on the stack, but for TRANSFORM-SET there is an option (using 2; instead of ;) when the word leaves two non negative integers on the stack

{ 0 10 | all }

creates the set {0,1,2,3,4,5,6,7,8,9} which can be transformed to a two dimensional set

:| n | n n ^2 2; transform-set cr zet.
{(0,0),(1,1),(2,4),(3,9),(4,16),(5,25),(6,36),(7,49),(8,64),(9,81)} ok

The purpose with this words is to be able to quickly inspect conjectures and Diophantine equations.

a and b are coprime if there exists values x and y such that

ax+by=1

Since BigZ just deals with sets of non negative numbers we can search x and y in the equation

ax-by=1

1 10 :| b a y x | a x * b y * - 1 = ; create-set cr zet.
{(2,1,1,1),(3,2,1,1),(4,3,1,1),(5,4,1,1),(6,5,1,1),(7,6,1,1),(8,7,1,1),(9,8,1,1),(1,1,2,1),(2,3,2,1),(3,5,2,1),(4,7,2,1),(5,9,2,1),(1,2,3,1),(2,5,3,1),(3,8,3,1),(1,3,4,1),(2,7,4,1),(1,4,5,1),(2,9,5,1),(1,5,6,1),(1,6,7,1),(1,7,8,1),(1,8,9,1),(3,1,1,2),(5,2,1,2),(7,3,1,2),(9,4,1,2),(1,1,3,2),(3,4,3,2),(5,7,3,2),(1,2,5,2),(3,7,5,2),(1,3,7,2),(1,4,9,2),(4,1,1,3),(7,2,1,3),(2,1,2,3),(5,3,2,3),(8,5,2,3),(1,1,4,3),(4,5,4,3),(7,9,4,3),(2,3,5,3),(5,8,5,3),(1,2,7,3),(4,9,7,3),(2,5,8,3),(5,1,1,4),(9,2,1,4),(3,2,3,4),(7,5,3,4),(1,1,5,4),(5,6,5,4),(3,5,7,4),(1,2,9,4),(6,1,1,5),(3,1,2,5),(8,3,2,5),(2,1,3,5),(7,4,3,5),(4,3,4,5),(9,7,4,5),(1,1,6,5),(6,7,6,5),(3,4,7,5),(2,3,8,5),(4,7,9,5),(7,1,1,6),(5,4,5,6),(1,1,7,6),(7,8,7,6),(8,1,1,7),(4,1,2,7),(5,2,3,7),(2,1,4,7),(9,5,4,7),(3,2,5,7),(6,5,6,7),(1,1,8,7),(8,9,8,7),(4,5,9,7),(9,1,1,8),(3,1,3,8),(5,3,5,8),(7,6,7,8),(1,1,9,8),(5,1,2,9),(7,3,4,9),(2,1,5,9),(4,3,7,9),(8,7,8,9)} ok

The vectors in the set are of the form (x,y,a,b), that is the opposite of its appearance as parameters. (Which is logical in Forth with postfix notations and values on a stack, but still a bit awkward).

The idea is a one row code and for that purpose there is a need of short words:

: isp  isprime ;         \ n -- flag
: isq  sqr ;             \ is perfect square: n -- flag
: isqf  sqrfree ;        \ is square free: n -- flag
: isem  bigomega 2 = ;   \ is semi prime: n -- flag
: ispp  smallomega 1 = ; \ is prime power: n -- flag

: 2sqs  2sqsum ;         \ square sum: a b -- sum
: 3sqs  3sqsum ;         \ square sum: a b c -- sum
: 4sqs  4sqsum ;         \ square sum: a b c d -- sum

: cop  coprime ;         \ are coprime: m n -- flag
: div  swap mod 0= ;     \ divides: m n -- flag

: <<  \ i j k -- flag  i<j<k
over > -rot < and ;

: <<=  \ i j k -- flag  i<=j<=k
over >= -rot <= and ;

: z. zet. ;

: fi postpone else postpone false postpone then ; immediate
\ a short version of ELSE FALSE THEN.

When inspecting a Diophantine equation there might be symmetries and trivial cases to weed out.

1 100 :| c b a | a b c << a b cop and if a b 2sqs c ^2 = else false then ; create-set

can be shortened to

1 100 :| c b a | a b c << a b cop and if a b 2sqs c ^2 = fi ; create-set

cr zet.
{(3,4,5),(5,12,13),(8,15,17),(7,24,25),(20,21,29),(12,35,37),(9,40,41),(28,45,53),(11,60,61),(33,56,65),(16,63,65),(48,55,73),(36,77,85),(13,84,85),(39,80,89),(65,72,97)} ok

Experimenting with sets gives a great opportunity to find and test conjectures:

1 50 :| b a | a b 2sqsum isprime ; create-set  ok
:| b a | a b + ; transform-set cr z.
{2,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,83,85,87,89,91,93,95} ok

This suggests that all odd numbers larger than 1 can be written as a sum a+b where a²+b² is prime.

Define

: sqeq \ m n a b -- ma²+nb²
dup * rot * -rot
dup * * + ;

and the set {a+b<50|a,b>0 & ma²+nb² is prime} for some different values of m and n

5 value m
7 value n

1 50 :| b a | m n a b sqeq isprime ; create-set
:| b a | a b + ; transform-set cr zet.
{5,7,11,13,17,19,23,25,29,31,35,37,41,43,47,49,53,55,59,61,65,67,71,73,77,79,83,85,89,91} ok

3 to m
4 to n

1 50 :| b a | m n a b sqeq isprime ; create-set
:| b a | a b + ; transform-set cr zet.
{2,3,4,5,6,8,9,10,11,12,13,15,16,17,18,19,20,22,23,24,25,26,27,29,30,31,32,33,34,36,37,38,39,40,41,43,44,45,46,47,48,50,51,52,53,54,55,57,58,59,60,61,62,64,65,66,67,68,69,71,72,73,74,75,76,78,79,80,81,82,83,86,87,88,89,93,94,95,96} ok

This will show a pattern which suggest that
{a+b|a,b>0 & ma²+nb² is prime}={k>1|gcd(k,m+n)=1}

However, further tests will show that some changes are needed:

Routines to check the conjecture and print the table:

: prime_partition \ m n k -- flag
false locals| flag k |
k 1
?do 2dup k i - i sqeq isprime
if true to flag leave then
loop 2drop flag ;

\ 100000 value nx     Gives the same table as
1000 value nx

:
maximal_exception \ m n -- mx
2dup + 1 locals| mx m+n n m |
nx 3
do m+n i coprime
if m n i
prime_partition 0=
if i to mx then
then
loop mx ;

: table \ m2 n2 --
locals| n2 | cr
2 spaces dup 1 do i 3 .r loop 1
do n2 1 cr i 2 .r
do j i coprime
if j i
maximal_exception
else 0
then 3 .r
loop
loop ;

The source code for BigZ can be loaded here:

https://github.com/Lehs/BigZ/blob/master/bigzet.txt

## Tuesday, May 30, 2017

### Karatsuba multiplication

The time for direct multiplication is proportional to n² where n is the number of figures in the multiplicands x,y. When choosing a big base B=2ⁿ and and writing

xy=(x0+B*x1)(y0+B*y1)=x0y0 + (x0*y1+x1*y0)B + x1*y1*B²

there are four multiplications of smaller numbers, also here the calculation time is proportional to n². However

(x0*y1+x1*y0)=(x0+x1)(y0+y1)-x0*y0-x1*y1

why it's enough to calculate three smaller multiplications

x0*y0, x1*y1 and (x0+x1)(y0+y1).

The multiplication with B are fast left shifting and if the shifting and the addition where cost free, the recursive Karatsuba multiplication would be very efficient

but unfortunately the extra math (and in Forth also some stack juggling) takes a lot of time and the method is efficient only for rather big numbers. For very big numbers, however, it's very efficient.

Here is the way I implemented it in ANS Forth:

: bcells* \ big m -- big*C^m
cells top\$ locals| n ad mb |
mb bvp @ +! ;
\ C is the number of digits in a cell

: bcells/ \ big m -- big/C^m
cells top\$ locals| n ad mb |
mb negate bvp @ +! ;

: bsplit \ w ad -- u v
dup nextfree <
if bvp @ dup @ vp+ bvp @ ! !
else drop bzero
then ;
\ A big number is split on the big stack at address ad

: btransmul \ x y -- x0 x1 y0 y1 m     B=2^bits
len1 len2 max cell + lcell 1+ rshift     \ m
dup >r cells
>bx first over + bsplit
bx> first + bsplit r> ;
\ x=x0+x1*B^m  y=y0+y1*B^m

0x84 value karalim \ break point byte length for termination.

: b* \ x y -- xy
len1 len2 max karalim <
if b* exit then
btransmul >r                   \ x0 x1 y0 y1
3 bpick 2 bpick recurse >bx    \ bx: x0*y0
2 bpick 1 bpick recurse >bx    \ bx: x0*y0 x1*y1
b+ >bx b+ bx>   recurse        \ (x0+x1)(y0+y1)
bx b- by b- r@ bcells*         \ z1*C^m
bx> r> 2* bcells* bx> b+ b+ <top ;
\ Karatsuba multiplication

## 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 ;

ok

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

Erdős 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

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 ;

## Saturday, March 18, 2017

### How to use BigZ - part 2

Gaps in increasing sequences of natural numbers

Strictly increasing sequences as

(1,2,3,...), (2,3,5,7,11,13,...) and (1,2,4,8,16,...)

can partly be represented as sorted sets in BigZ. Differences of sequences is an analogy to differentiation of functions. When defining a function gapz, to be the sorted set of all gaps between consecutive numbers of a set, also gapz become an analogy of the derivative of functions.

Apply gapz on an arithmetic sequence gives a set with a single element. If n times apply gapz on an increasing polynomial series of degree n gives a singleton set. On a sequence of exponential function with base two does nothing to the infinite set.

: gapz \ s -- s'
0 locals| n |        \ counts the number of gaps in s'
foreach 1+           \ prepare elements of s for the do-loop
do zst> zst@ - >xst  \ the gap between the largest consecutive's
n 1+ to n
loop zst> drop       \ drop the smallest element of s
n 2* negate >xst     \ calculate the set-count for s'
xst zst setmove      \ move the set to zst
set-sort reduce ;    \ sort and eliminate copies

{ 1 1000 | prime } gapz cr zet.

{1,2,4,6,8,10,12,14,18,20} ok

Partitions of a number n into distinct primes

: collincl \ s n -- s'
0 >xst
begin zst@
while zsplit
dup >zst zfence zmerge
set-sort reduce zfence
xst zst setmove zmerge
zst xst setmove
repeat zdrop drop
xst zst setmove
reduce ;
\ include n in all sets in s

: xunion \ set --
xst zst setmove union
zst xst setmove ;
\ Union of the top sets on the xst- and zst-stacks
\ is put on the xst-stack

: primeset \ m -- set
pi dup 1+ 1
?do i pnr@ >zst
loop 2* negate >zst ;
\ Create the set of all primes < m+1

: memb \ s n -- flag
false swap
?do dup i @ =
if -1 under+ leave then cell
+loop drop ;
\ Faster test if n is a member in the sorted number set s

For T being the set of primes:

The algorithm can be used with corrections for n=2p.

: termcase \ n -- flag
case 2 of true endof
3 of true endof
11 of true endof
dup of false endof
endcase ;
\ terminal cases: prime numbers without additional partitions

I have no proof that there are additional partitions for all primes greater than 11, but as far as the algorithm will go the terminal cases above are correct.

: z2@ \ set -- set n
zst> zst@ swap >zst ;
\ read the largest element in the set

: lowlim \ set n -- set p
0 swap adn1 over + swap
?do i @ under+ 2dup < 0=
if 2drop i @ leave then cell
+loop ;
\ p is the smallest prime such that 2+3+5+...+p > n

: setsum \ set -- sum
0 foreach ?do zst> + loop ;
\ The sum of all elements in set

: sumcorr \ s n -- s'
locals| n |
0 >xst
begin zst@
while zsplit zdup setsum n =
if zfence xunion
else zdrop
then
repeat zst> drop
xst zst setmove ;
\ Removes all partitions from s such that the sum < n

: dps \ n -- set
dup 2 < if drop 0 >zst exit then
dup termcase if >zst -2 >zst -4 >zst exit then
0 >xst
dup primeset
dup lowlim locals| low n |
begin zst@
if z2@ low <
if false else true then
else false
then
while zsplit n zst> dup >r - ?dup
if recurse
zst@
if r> collincl n sumcorr xunion
else zst> drop r> drop
then
else { { r> } } xunion
then
repeat zdrop
xst zst setmove

set-sort reduce ;
\ The set of partitions of n>0 into distinct primes

20 dps cr zet.

{{2,7,11},{2,5,13},{7,13},{3,17}} ok

50 dps cr zet.
{{2,7,11,13,17},{2,5,11,13,19},{7,11,13,19},{2,5,7,17,19},{3,11,17,19},{2,5,7,13,23},{3,11,13,23},{2,3,5,17,23},{3,7,17,23},{3,5,19,23},{2,3,5,11,29},{3,7,11,29},{3,5,13,29},{2,19,29},{3,5,11,31},{2,17,31},{19,31},{2,11,37},{13,37},{2,7,41},{2,5,43},{7,43},{3,47}} ok

: A000586 \ n --
." 1," 1+ 1
?do i dps cardinality 0
<# [char] , hold #s #> type
loop ;
\ List A000586

100 a000586 cr
1,0,1,1,0,2,0,2,1,1,2,1,2,2,2,2,3,2,4,3,4,4,4,5,5,5,6,5,6,7,6,9,7,9,9,9,11,11,11,13,12,14,15,15,17,16,18,19,20,21,23,22,25,26,27,30,29,32,32,35,37,39,40,42,44,45,50,50,53,55,57,61,64,67,70,71,76,78,83,87,89,93,96,102,106,111,114,119,122,130,136,140,147,150,156,164,170,178,183,188,198, ok

Partitions of a number n into distinct non composites

A variant of the above.

: termcase1 \ n -- flag
case 1 of true endof
2 of true endof
dup of false endof
endcase ;

: dps1 \ n -- set
dup 0= if >zst exit then
dup termcase1 if >zst -2 >zst -4 >zst exit then
0 >xst
dup { 1 } primeset zmerge
dup lowlim locals| low n |
begin zst@
if z2@ low <
if false else true then
else false
then
while zsplit n zst> dup >r - ?dup
if recurse
zst@
if r> collincl n sumcorr xunion
else zst> drop r> drop
then
else { { r> } } xunion
then
repeat zdrop
xst zst setmove
set-sort reduce ;

50 dps1 cr zet.
{{1,3,5,11,13,17},{2,7,11,13,17},{1,2,3,5,7,13,19},{2,5,11,13,19},{7,11,13,19},{2,5,7,17,19},{1,2,11,17,19},{3,11,17,19},{1,13,17,19},{1,3,5,7,11,23},{2,5,7,13,23},{1,2,11,13,23},{3,11,13,23},{2,3,5,17,23},{1,2,7,17,23},{3,7,17,23},{1,2,5,19,23},{3,5,19,23},{1,7,19,23},{2,3,5,11,29},{1,2,7,11,29},{3,7,11,29},{1,2,5,13,29},{3,5,13,29},{1,7,13,29},{1,3,17,29},{2,19,29},{1,2,5,11,31},{3,5,11,31},{1,7,11,31},{1,2,3,13,31},{1,5,13,31},{2,17,31},{19,31},{1,2,3,7,37},{1,5,7,37},{2,11,37},{13,37},{1,3,5,41},{2,7,41},{2,5,43},{7,43},{1,2,47},{3,47}} ok

: test \ n --    n>0
1+ 1
?do i dps1 cardinality 0
<# [char] , hold #s #> type
loop ;

100 cr test
1,1,2,1,2,2,2,3,2,3,3,3,4,4,4,5,5,6,7,7,8,8,9,10,10,11,11,11,13,13,15,16,16,18,18,20,22,22,24,25,26,29,30,32,33,34,37,39,41,44,45,47,51,53,57,59,61,64,67,72,76,79,82,86,89,95,100,103,108,112,118,125,131,137,141,147,154,161,170,176,182,189,198,208,217,225,233,241,252,266,276,287,297,306,320,334,348,361,371,386, ok