15-page article belonging to my "Boldly Going" series, this is the original version for the HP-71B of my article "HP Article VA042a - Boldly Going - Outsmarting PROOT (42S)".

Though this article appears at first sight quite similar to the version for the HP-42S published in Update #24, it was actually written earlier and includes additional sections discussing the internal workings of PROOT and their differences, as well as more examples specific to the 12-digit precision available in the HP-71B, which aren't featured in the HP-42S version. Also, it can be very useful for HP-42S users to better understand the RPN version innards, as the BASIC code is much clearer and the algorithm is exactly the same.

The article includes a small 11-line BASIC subprogram PZER (plus an optional driver) for the HP-71B and emulators (Emu71/DOS, Emu71/Win, go71b, etc.) dealing with a glaring limitation of the HP-71B Math ROM’s keyword PROOT, the polynomial root finder which finds at once all real/complex roots of an Nth-degree polynomial but only works for polynomials having real coefficients. On the other hand, PZER works efficiently and accurately for polynomials whose coefficients are real and/or complex, and can work globally (no user inputs or initial guesses required whatsoever, just the coefficients) while also providing the user with extra control if desired. It can be called from your own programs or manually from the command line.

On-line (i.e. stream) algorithm for the calculation of approximate quantiles.
Found via https://www.hpmuseum.org/forum/thread-24780-post-217334.html#pid217334 →
https://math.stackexchange.com/questions/4848507/which-2nd-root-finding-method-does-the-hp32e-use-to-compute-q-1

Emacs Calc, with different precision p (decimal digits):
        p       sin(1e30°)
   [ [ 99  -0.984807753012 ]
     [ 50  -0.984807753012 ]
     [ 40  -0.984807753012 ]
     [ 35  -0.984807753982 ]
     [ 30  -0.984794189056 ]
     [ 20        0.        ] ]

hp33s
-0.984 807 753 012
free42
-0.984 807 753 012 208 059 366 743 024 589 523
jrpn15c
-0.984 807 753 012 208 059 366 743 024 589 523

Found on the MoHP forum https://www.hpmuseum.org/forum/thread-23365-post-202412.html#pid202412
See also the interesting http://www.luschny.de/math/zeta/The-Bernoulli-Manifesto.html which convinced Donald Knuth to correct the definition of the Bernoulli number B_1 in his book Concrete Mathematics.
The index http://www.luschny.de/math/index.htm gives a good overview (in german).

A source for the statement "The actual number of gates needed to factor a
n-bit number is 72 * n^3; so for 15, it's 4 bits, 4608 gates; not happening any
time soon." would be nice.

"How do you fit a dictionary in 64kb RAM? Unix engineers solved it with clever data structures and compression tricks. Here's the fascinating story behind it."
Mentioned by Aharon Robbins on the TUHS mailing list https://www.tuhs.org/pipermail/tuhs/2025-January/031371.html

Includes a link to https://technical.swissmicros.com/doc/pdf/Bairstow.pdf

Lecture notes by Dr. Madhu Sudan, found on Wikipedia https://en.wikipedia.org/wiki/Berlekamp%E2%80%93Welch_algorithm

Uses Bairstow's algorithm, code for dm42 and a PDF document. A matlab version is mentioned.

primecount is a command-line program and C/C++ library that counts the number of primes ≤ x (maximum 1031) using highly optimized implementations of the combinatorial prime counting algorithms.
Found via https://www.cliki.net/primecount and https://github.com/AaronChen0/primecount