The Gilbreath’s Conjecture

June 12, 2014

(This article was first published on Ripples, and kindly contributed to R-bloggers)

317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way (G.H. Hardy)

In 1958, the mathematician and magician Norman L. Gilbreath presented a disconcerting hypothesis conceived in the back of a napkin. Gilbreath wrote first prime numbers in a row. In the next rows, he wrote the difference  in absolute value of consecutive values of previous row. Beginning with first 20 primes in the first row, he obtained something like this:Gilbreath01

The conjecture is easy: except for the first one, all elements of the first column are 1. So far, no one has demonstrated this hypothesis. In fact, according to mathematician Richard Guy, it seems unlikely that we will see a demonstration of Gilbreath’s conjecture in the near future, though probably this conjecture is true.

In the previous chart, I coloured zeros in white, ones in violet and rest of numbers in gold. The conjecture says that except for the first element, the first column is entirely violet. Following you can see the coloured chart for first 20, 40, 60 and 80 prime numbers:


It is nice how zeros create triangular patterns similar to patterns created by cellular automata. This is the chart for first 200 primes:Gilbreath03

How much time will it take to demonstrate this simple conjecture? Who knows. Meanwhile, you can draw triangles with this code:

gilbreath=t(matrix(primlist[2:(n+1)], n, n))
for (i in 2:n) {gilbreath[i,]=c(eval(parse(text=paste(c(paste("abs(diff(",collapse=""), "gilbreath[i-1,]", paste("))",collapse="")),collapse=""))),rep(NA,1))}
opt <- theme(legend.position="none",
panel.background = element_blank(),
panel.grid = element_blank(),
axis.text =element_blank())
gilbreath$value1=cut(gilbreath$value, breaks=c(-Inf,1:2,Inf), right = FALSE)
ggplot(gilbreath, aes(x=X1, y=X2)) +
geom_tile(aes(fill = value1), colour="grey") +
scale_fill_manual(values = c("white", "darkviolet", "gold"))+
geom_text(label=gilbreath$value, size=8)+

To leave a comment for the author, please follow the link and comment on their blog: Ripples. offers daily e-mail updates about R news and tutorials on topics such as: Data science, Big Data, R jobs, visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, git, hadoop, Web Scraping) statistics (regression, PCA, time series, trading) and more...

If you got this far, why not subscribe for updates from the site? Choose your flavor: e-mail, twitter, RSS, or facebook...

Comments are closed.


Mango solutions

RStudio homepage

Zero Inflated Models and Generalized Linear Mixed Models with R

Quantide: statistical consulting and training


CRC R books series

Contact us if you wish to help support R-bloggers, and place your banner here.

Never miss an update!
Subscribe to R-bloggers to receive
e-mails with the latest R posts.
(You will not see this message again.)

Click here to close (This popup will not appear again)