Category Archives: Conjectures

The Andrica’s Conjecture

Conjectures, , , , , ,

Things should be as simple as possible, but not simpler (Albert Einstein)

Following with conjectures about primes, it is time for Andrica’s conjecture. The great mathematician Leonhard Euler (1707-1783) pointed: “Mathematicians have tried with no success to find some kind of order in the sequence of prime numbers and today we have reasons to believe that this is a mystery that human mind will never understand”.

In 1985, the Romanian mathematician Dorin Andrica published his conjecture, still unproved, which makes reference to gap between consecutive prime numbers. In concrete, his conjecture establishes that difference between square roots between two consecutive prime numbers is always less than 1. The highest difference encountered until now is 0.67087, located between p4=7 and p5=11.

Following you can find the plot of these differences for first 400 prime numbers:

andricaIt is very interesting how dots form hyperbolic patterns. Does not seem similar in some sense to the Ulam spiral? Primes: how challenging you are!

Two more comments:

  • It is better to find primes using matlab package than doing with schoolmath one. Reason is simple: for schoolmath package, 133 is prime!
  • Why did Andrica formulated his conjecture as √pn+1-√pn < 1 instead of √pn+1-√pn < 3/4? In terms of statistical error, the second formulation is more accurate. Maybe the charisma of number 1 is hard to avoid.

This is the code. I learned how to insert mathematical expressions inside a ggplot chart:

library(matlab)
library(ggplot2)
ubound=2800
primes=primes(ubound)
andrica=data.frame(X=seq(1:(length(primes)-1)), Y=diff(sqrt(primes)))
opt=theme(panel.background = element_rect(fill="gray92"),
panel.grid.minor = element_blank(),
panel.grid.major.x = element_blank(),
panel.grid.major.y = element_line(color="white", size=1.5),
plot.title = element_text(size = 45),
axis.title = element_text(size = 28, color="gray35"),
axis.text = element_text(size=16),
axis.ticks = element_blank(),
axis.line = element_line(colour = "white"))
ggplot(andrica, aes(X, Y, colour=Y))+geom_point(size=5, alpha=.75)+
scale_colour_continuous(guide = FALSE)+
scale_x_continuous("n", limits=c(0, length(primes)-1), breaks = seq(0,length(primes)-1,50))+
scale_y_continuous(expression(A[n]==sqrt(p[n+1])-sqrt(p[n])), limits=c(0, .75), breaks = seq(0,.75,.05))+
labs(title = "The Andrica's Conjecture")+
opt

The Goldbach’s Comet

Conjectures, ,

Every even integer greater than 2 can be expressed as the sum of two primes (Christian Goldbach, 1742)

The point cloud known as Goldbach’s Comet represents the amount of different ways (y axis) an even number (x axis) can be writen as sum of two prime numbers. In this plot, x axis is between 2 and 50.000 (stars are just ornaments):

Goldbach

Mathematicians are still waiting for a proof of this conjecture. This is the code for drawing this plot:

require(schoolmath)
require(utils)
library(plyr)
n=50000
data(primlist)
primes=as.data.frame(t(combn(primlist[primlist>2 & primlist<n-2], 2)))
primes$V3=primes$V1+primes$V2
primes2=count(primes, "V3")
primes2=primes2[primes2$V3<=n,]
stars=cbind(runif(50, min=-n*0.05, max=n), runif(200, min=-n*0.001, max=max(primes2$freq)))
plot.new()
par(mai = rep(0, 4), bg = "gray12")
plot(NA,type="n", xlim=c(-n*0.05,n), xaxs="i", ylim=c(-n*0.001,max(primes2$freq)))
points(stars, col = "blue4", cex=.7, pch=16)
points(stars, col = "blue", cex=.3, pch=16)
points(stars, col = "gray75", cex=.1, pch=16)
apply(primes2, 1, function(x) points(x=x[1],y=x[2], col = if (runif(1)&gt;x[1]/n) {"white"} else {sample(colours(),1)}, cex=.1, pch=16))