golden bug [jatp]

genuine Latinsquare rectangle

the strange occurrence of the one bump

When answering an X validated question on running an accept-reject algorithm for the Gamma distribution by using a mixture of Beta and drifted (bt 1) Exponential distributions, I came across the above glitch in the fit of my 10⁷ simulated sample to the target, apparently displaying a wrong proportion of simulations above (or below) one.

a=.9
g<-function(T){
  x=rexp(T)
  v=rt(T,1)<0
  x=c(1+x[v],exp(-x/a)[!v])
  x[runif(T)<x^a/x/exp(x)/((x>1)*exp(1-x)+a*(x<1)*x^a/x)*a]}

It took me a while to spot the issue, namely that the output of

  z=g(T)
  while(sum(!!z)<T)z=c(z,g(T))
  z[1:T]

was favouring simulations from the drifted exponential by truncating. Permuting the elements of z before returning solved the issue (as shown below for a=½)!

weird bug

Le Monde puzzle [#6]

A simple challenge in Le Monde this week: find the group of four primes such that any sum of three terms in the group is prime and the overall sum is minimised. Here is a quick exploration by simulation, using the schoolmath package (with its imperfections):

A=primes(start=1,end=53)[-1]
lengthA=length(A)

res=4*53
for (t in 1:10^4){

 B=sample(A,4,prob=1/(1:lengthA))
 sto=is.prim(sum(B[-1]))
 for (j in 2:4)
 sto=sto*is.prim(sum(B[-j]))

 if ((sto)&(sum(B)<res)){
 res=sum(B)
 sol=B}
 }
}

providing the solution 5 7 17 19.

A subsidiary question in the same puzzle is whether or not it is possible to find a group of five primes such that any sum of three terms is still prime. Running the above program with the proper substitutions of 4 by 5 does not produce any solution, even when increasing the upper boundary in A. So it is most likely that the answer is no.