POBA: Primes Only Best Approximate

It takes a little getting used to the concept of what a POBA or Primes Only Best Approximate is. One definition runs:

"Suppose that \( x > 0 \). A fraction \( p/q \) of primes is a primes-only best approximate (POBA), and we write "\( p/q \) in B( \( x \) )", if \(0 < |x - p/q| < |x - u/v| \) for all primes \(u\) and \(v\) such that \(v < q\), and also, \( |x - p/q| < |x - p'/q|  \) for every prime \(p'\) except \(p\). Note that for some choices of \(x\), there are values of \(q\) for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...)." 

Things become clearer if we look at a specific example. On April 23rd 2022, I turned 26683 days old and one of the properties of this number is that it's a member of OEIS A265788:

A265788

Numerators of primes-only best approximates (POBAs) to \( \sqrt{5} \).

The POBAs to start with are 3/2, 5/2, 7/3, 11/5, 29/13, 163/73, 199/89, 521/233. The initial members of the sequence are thus 3, 5, 7, 11, 29, 163, 199, 521 (with the continuation being 3571, 26683, 111667, 150427, 154841 etc.). Table 1 shows the initial approximations:

Table 1: permalink POBAs for \( \sqrt{5}\)

The convergence is far slower than with the continued fraction  where the numerator and denominator of each fraction are coprime. See Table 2 and note that scientific notation is used once the difference becomes very small.

Table 2: permalink continued fraction approximations for \( \sqrt{5} \)

The SageMath algorithm I developed is rather sluggish but it can be easily modified to accommodate other irrational or transcendental numbers such as \( \pi \). See Table 3 for the POBAs for \( \pi \).

Table 3: permalink POBAs for \( \pi \)

Once again, the continued fractions are far more efficient and the display quickly changes to scientific notation because of the small differences. See Table 4 for the continued fraction approximations of \( \pi \).

Table 4: permalink continued fraction approximations for \( \pi \)

Let's not forget that the continued fractions for irrational and transcendental numbers offer the best approximations. While the POBAs are clearly less efficient, they are nonetheless interesting in their own right. Further variations can be envisaged such as fractions whose numerators and denominators consist of only Fibonacci numbers rather than primes.