The Universe of Discourse

In which I revisit the pastimes of my misspent youth
Last weekend I was at a flea market and saw an HP-15C calculator for $10. The HP-15C was the last pocket calculator I owned, some time before pocket calculators became ridiculous. It was a really nice calculator when I got it in 1986, one of my most prized possessions.

Then on Friday night I was sitting around thinking about which numbers n are such that !!10n^2+9!! a perfect square, and I couldn't think of any examples except for 0, 2, and 4. Normally I would just run and ask the computer, which would take about two minutes to write the program and one second to run it. But I was out in the courtyard, it was a really nice evening, my favorite time of the year, the fading light was beautiful, and I wasn't going to squander it by going inside to brute-force some number problem.

But I did have the HP-15C in my pocket, and the HP-15C is programmable, by mid-1980s programmable calculator standards. That is to say, it is just barely programmable, but just barely is all you need to implement linear search for solutions of !!10n^2+9 = m^2!!. So I wrote the program and discovered, to my surprise, that I still remember many of the fussy details of how to program an HP-15C. For example, the SST button single-steps through the listing, in program mode, but single-steps the execution in run mode. And instead of using the special test 5 to see if the x and y registers are equal you might as well subtract them and use the x=0 test; it uses the same amount of program memory and you won't have to flip the calculator over to remember what test 5 is. And the x² and INT() operations are on the blue shift key.

Here's the program:

        001 - 42,21,11      Label A:  (subroutine)
        002 -    43 11        x²
        003 -        1
        004 -        0        10
        005 -       20        multiply
        006 -        9        9
        007 -       40        add
        008 -       36        enter (dup)
        009 -       11        √
        010 -       36        enter (dup)
        011 -    43 44        x ← int(x) 
        012 -       30        subtract
        013 -    43 20        unless x=0:
        014 -       31          STOP
        015 -    43 32        return from subroutine
        016 - 42,21,12      Label B:
        017 -       40        +
        018 -    45  0        load register 0
        019 -    32 11        call A
        020 -        2        2
        021 - 44,40, 0        add to register 0
        022 -    22 12        goto B

        010 -       36   -- enter (dup)
        011 -    43 44   -- x ← INT(x)
        012 -       30   -- subtract
        013 -    43 20   -- unless x=0: …

                 42 44   -- x ← FRAC(x)
                 43 20   -- unless x=0: …

Besides, stuff should be done right. That's why it's called "right".

But I kind of wish I had that part of my brain back. Who knows what useful thing I would be able to remember if I wasn't wasting my precious few brain cells remembering that the back-step key ("BST") is on the blue shift, and that "42,21,12" is the code for "subroutine B starts here".

Anyway, the program worked, once I had debugged it, and in short order (by 1986 standards) produced the solutions n=18, 80, 154, which was enough to get my phone to search the OEIS and find the rest of the sequence. The OEIS entry mentioned that the solutions have the generating function

$$\frac{2x^2(1+2x+9x^2+2x^3+x^4)}{1-38x^3+x^6}$$

[ Addendum 20200204: Had I been doing this in the 1980s, I would have had to go into the house to check the OEIS, but I have wistfully gotten rid of my useless hardback copy, because these days I can do it from the courtyard on my phone. ]

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