Square form factorization
A just for fun snail ..
# 20210325 Raku programming solution
my @multiplier = ( 1, 3, 5, 7, 11, 3*5, 3*7, 3*11, 5*7, 5*11, 7*11, 3*5*7, 3*5*11, 3*7*11, 5*7*11, 3*5*7*11 );
sub circumfix:<⌊ ⌋>{ $^n.floor }; sub prefix:<√>{ $^n.sqrt }; # just for fun
sub SQUFOF ( \𝑁 ) {
return 1 if 𝑁.is-prime; # if n is prime return 1
return √𝑁 if √𝑁 == Int(√𝑁); # if n is a perfect square return √𝑁
for @multiplier -> \𝑘 {
my \Pₒ = $ = ⌊ √(𝑘*𝑁) ⌋; # P[0]=floor(√N)
my \Qₒ = $ = 1 ; # Q[0]=1
my \Q = $ = 𝑘*𝑁 - Pₒ²; # Q[1]=N-P[0]^2 & Q[i]
my \Pₚᵣₑᵥ = $ = Pₒ; # P[i-1] = P[0]
my \Qₚᵣₑᵥ = $ = Qₒ; # Q[i-1] = Q[0]
my \P = $ = 0; # P[i]
my \Qₙₑₓₜ = $ = 0; # P[i+1]
my \b = $ = 0; # b[i]
# i = 1
repeat until √Q == Int(√Q) { # until Q[i] is a perfect square
b = ⌊⌊ √(𝑘*𝑁) + Pₚᵣₑᵥ ⌋ / Q ⌋; # floor(floor(√N+P[i-1])/Q[i])
P = b*Q - Pₚᵣₑᵥ; # P[i]=b*Q[i]-P[i-1]
Qₙₑₓₜ = Qₚᵣₑᵥ + b*(Pₚᵣₑᵥ - P); # Q[i+1]=Q[i-1]+b(P[i-1]-P[i])
( Qₚᵣₑᵥ, Q, Pₚᵣₑᵥ ) = Q, Qₙₑₓₜ, P; # i++
}
b = ⌊ ⌊ √(𝑘*𝑁)+P ⌋ / Q ⌋; # b=floor((floor(√N)+P[i])/Q[0])
Pₚᵣₑᵥ = b*Qₒ - P; # P[i-1]=b*Q[0]-P[i]
Q = ( 𝑘*𝑁 - Pₚᵣₑᵥ² )/Qₒ; # Q[1]=(N-P[0]^2)/Q[0] & Q[i]
Qₚᵣₑᵥ = Qₒ; # Q[i-1] = Q[0]
# i = 1
loop { # repeat
b = ⌊ ⌊ √(𝑘*𝑁)+Pₚᵣₑᵥ ⌋ / Q ⌋; # b=floor(floor(√N)+P[i-1])/Q[i])
P = b*Q - Pₚᵣₑᵥ; # P[i]=b*Q[i]-P[i-1]
Qₙₑₓₜ = Qₚᵣₑᵥ + b*(Pₚᵣₑᵥ - P); # Q[i+1]=Q[i-1]+b(P[i-1]-P[i])
last if (P == Pₚᵣₑᵥ); # until P[i+1]=P[i]
( Qₚᵣₑᵥ, Q, Pₚᵣₑᵥ ) = Q, Qₙₑₓₜ, P; # i++
}
given 𝑁 gcd P { return $_ if $_ != 1|𝑁 }
} # gcd(N,P[i]) (if != 1 or N) is a factor of N, otherwise try next k
return 0 # give up
}
race for (
11111, # wikipedia.org/wiki/Shanks%27s_square_forms_factorization#Example
4558849, # example from talk page
# all of the rest are taken from the FreeBASIC entry
2501,12851,13289,75301,120787,967009,997417,7091569,13290059,
42854447,223553581,2027651281,11111111111,100895598169,1002742628021,
# time hoarders
60012462237239, # = 6862753 * 8744663 15s
287129523414791, # = 6059887 * 47381993 80s
11111111111111111, # = 2071723 * 5363222357 2m
384307168202281507, # = 415718707 * 924440401 5m
1000000000000000127, # = 111756107 * 8948056861 12m
9007199254740931, # = 10624181 * 847801751 17m
922337203685477563, # = 110075821 * 8379108103 41m
314159265358979323, # = 317213509 * 990371647 61m
1152921505680588799, # = 139001459 * 8294312261 93m
658812288346769681, # = 62222119 * 10588072199 112m
419244183493398773, # = 48009977 * 8732438749 135m
1537228672809128917, # = 26675843 * 57626245319 254m
# don't know how to handle this one
# for 1e-323, 1e-324 { my $*TOLERANCE = $_ ;
# say 4611686018427387877.sqrt ≅ 4611686018427387877.sqrt.Int }
# skip the perfect square check and start k with 3 to get the following
# 4611686018427387877, # = 343242169 * 13435662733 217m
) -> \data {
given data.&SQUFOF {
when 0 { say "The number ", data, " is not factored." }
when 1 { say "The number ", data, " is a prime." }
default { say data, " = ", $_, " * ", data div $_.Int }
}
}Output:
Use NativeCall
Output:
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