https://www.johndcook.com/blog/2025/02/18/the-buenos-aires-constant/

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The Buenos Aires constant

Posted on 18 February 2025 by John

The Buenos Aires constant is 2.92005097731613...

What's so special about this number? Let's see when we use it to
initialize the following Python script.

s = 2.920050977316134

for _ in range(10):
    i = int(s)
    print(i)
    s = i*(1 + s - i)

What does this print?

    2, 3, 5, 7, 11, 13, 17, 19, 23, 29

If we started with the exact Buenos Aires constant and carried out
our operations exactly, the procedure above would generate all prime
numbers. In actual IEEE 754 arithmetic, it breaks down around Douglas
Adams' favorite number.

The Buenos Aires constant is defined by the infinite sum

\lambda = \frac{2-1}{1} + \frac{3-1}{2} + \frac{5-1}{3\cdot 2} + \
frac{7-1}{5\cdot 3 \cdot 2} + \cdots

As you can tell, the primes are baked into the definition of l, so
the series can't generate new primes.

I used mpmath to calculate l to 100 decimal places:

2.920050977316134712092562917112019468002727899321426719772682533107733772127766124190178112317583742

This required carrying out the series defining l for the first 56
primes. When I carried out the iteration above also to 100 decimal
places, it failed on the 55th prime. So I got about as many primes
out of the computation as I put into it.

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Reference: Beyond Pi and e: a Collection of Constants. James Grime,
Kevin Knudson, Pamela Pierce, Ellen Veomett, Glen Whitney. Math
Horizons, Vol. 29, No. 1 (September 2021), pp. 8-12

Categories : Math
Tags : Number theory
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2 thoughts on "The Buenos Aires constant"

 1.
    Sam
    18 February 2025 at 17:43

    If you write the loop as

    for _ in range(1, 11):

    It avoids the confusion of using i for two things. Or simpler

    for _ in range(10):

 2.
    Johnathan Corgan
    18 February 2025 at 19:33

    This reminds me of the compression scheme where you replace text
    with the offset into pi that contains the ascii encoding of the
    text...

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John D. Cook

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