# Complex Space Factorization

I have posted a discussion on Math.SE regarding visualizing the complex space and effects of factorization methods on that space. A quick link to the paper is available here. I have 3 routines in that paper, the first is the complex space generator. It works, but it is very slow and would appreciate any suggestions on improvement.

Complex Space Generator:

Complex.Space.Generator <- function(N){
min.real = ceiling(sqrt(N))
segtop_x = ceiling(N/6)+1
segtop_y = floor(N/6)-1
plot(N,N,xlim=c((0),N/3), ylim =c((0),N/6),xlab="Real",ylab="Imaginary")
segments(3,0,segtop_x,segtop_y)
segments(segtop_x,segtop_y,N/3,0, lty = "dotted")

for (i in 3:N/3){
for (j in 0:N/3){
i=as.integer(i)
j=as.integer(j)
if((i-j)>=3 && (i+j)<=(N/3) && (i-j)<=min.real)
points(i,j, pch = 17,col = ifelse((i-j)*(i+j) < N,'blue',ifelse((i-j)*(i+j) == N,'green','red')))
}
}
}


The second routine is the simultaneous factorization method involving Fermat and trial division. It is pretty fast, but I'm sure some improvements can be identified.

 Simultaneous.Complex.Factorization <- function(N){
min.real = ceiling(sqrt(N))
max.imaginary = min.real - 3
divisor = min.real + max.imaginary ### 135degree yellow line
### Estimate of where complex space turns red
min.imaginary = floor(mean(c((N/divisor),max.imaginary)))
### Corresponding real to above estimate
max.real = divisor - min.imaginary
j = 0L
while (j>=0L ){
if(
sqrt(((min.real + j)^2) - N)%%1==0 |   ### Vertical Fermat

sqrt(((max.imaginary - j)^2) + N)%%1==0 |    ### Horizontal Fermat

(N/((min.real+j)-(max.imaginary-j)))%%1==0 |    ### Trial Division

sqrt(((max.real - j)^2) - N)%%1==0 |    ### Vertical Fermat from estimate

(N/((max.real-j)-(min.imaginary+j)))%%1==0    ### Trial Division from estimate
) {

return(as.matrix(c(iterations=j,
mapped.real = (min.real+j)-(max.imaginary - j),
Factor_v = if((sqrt((min.real+j)^2 - N))%%1==0) (min.real+j)-sqrt((min.real+j)^2 - N),
Factor_h = if((sqrt((max.imaginary - j)^2 + N))%%1==0) (max.imaginary - j) + sqrt((max.imaginary - j)^2 + N),
Factor_td = if((N/((min.real+j)-(max.imaginary-j)))%%1==0) (N/((min.real+j)-(max.imaginary-j))),
Factor_v.2 = if((sqrt((max.real-j)^2 - N))%%1==0) (max.real-j)-sqrt((max.real-j)^2 - N),
Factor_td.2 = if((N/((max.real-j)-(min.imaginary+j)))%%1==0) (N/((max.real-j)-(min.imaginary+j)))
)))}

j = j + 1L

}
}


And finally I have the alternative method presented in the paper. It is very slow compared to the above method even though it requires noticeably less iterations.

Viole.Factorization <- function(N){
min.real = ceiling(sqrt(N))
max.imgainary = (N-9)/6
iterated.average = 0L

for (i in 2:log2(max.imgainary)){
iterated.average = mean(c(min.real,N))
iterated.average[i] = mean(c(iterated.average[i-1],N))
}

descending.iterations = floor(iterated.average)
ascending.iterations = ceiling(iterated.average)
print(iterated.average)
j = 0L

while(j>=0L){
for(i in 1:length(iterated.average)){
if( (descending.iterations[i]-j > descending.iterations[i-1] | ascending.iterations[i]+j < ascending.iterations[i+1]) &&
GCD(descending.iterations[i]-j,N)>1 && GCD(descending.iterations[i]-j,N)<N |
GCD(ascending.iterations[i]+j,N)>1 && GCD(ascending.iterations[i]+j,N)<N
){

return(c("Factor"=GCD(descending.iterations[i]-j,N),"Factor"=GCD(ascending.iterations[i]+j,N),"Iterated.Average.Level"=i,"Iterations"=j))}
}
j = j + 1L
}
}


I look forward to any insights regarding the routines. If you have any suggestions or questions on the content of the paper then please comment on the Math.SE discussion.