# Tupper’s self-referential formula

**StaTEAstics.**, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

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Can't remember where I first came across this equation but the Tupper's self referential equation, is a very interesting formula that when graphed in two dimension plane it reproduces the formula.

\[ \frac{1}{2} < \left\lfloor \bmod\left(\left\lfloor\frac{y}{17}\right\rfloor2^{-17\lfloor x\rfloor - \bmod(\lfloor y \rfloor, 17)}, 2\right)\right\rfloor \]

I first thought this would be a quick 5 min exercise which turned into a 3 hour work, the obstacle was that the constant “k” used in the formula is an extremely **big** integer and can not be handled in *R*naturally.

After a little search Large integer in R and play around it seems that the *gmp* library seems to work well.

First you will need to install **GMP** (The GNU Multiple Precision Arithmetic Library) from http://gmplib.org/, then install the **gmp**library within *R*.

## Load the library after installing GMP library(gmp) ## Define the constant k k = as.bigz("960939379918958884971672962127852754715004339660129306651505519271702802395266424689642842174350718121267153782770623355993237280874144307891325963941337723487857735749823926629715517173716995165232890538221612403238855866184013235585136048828693337902491454229288667081096184496091705183454067827731551705405381627380967602565625016981482083418783163849115590225610003652351370343874461848378737238198224849863465033159410054974700593138339226497249461751545728366702369745461014655997933798537483143786841806593422227898388722980000748404719") ## The tupper's formula tupper = function(x, y, k){ z1 = as.bigz(y + k) z2 = as.bigq(z1/17) z3 = 2^(-17 * x - as.bigz(z1%%17)) 0.5 < floor(as.bigz(z2 * z3)%%2) } ## The x and y axis x = 0:105 y = 0:16 ## Compute the matrix a = matrix(0, nc = length(x), nr = length(y)) for(i in seq_along(x)){ a[ ,107 - i] = rev(tupper(x[i], y, k = k)) }

Here is the plot

image(t(a), col = c("white", "black"))

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**StaTEAstics.**.

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