[VA] Short & Sweet Math Challenge #24: "2019 Spring Special 5-tier"

[Paul Dale]

I'm not sure why but tier 3 and tier 4 look like the easiest.

Spoilers towards the end.

Anyway, a 24 step program for tier 3 for the HP 25:

01 CLREG
02 1
03 x<>y
04 x=0?
05 GTO 24
06 2
07 /
08 INT
09 LASTx
10 FRAC
11 x=0?
12 GTO 17
13 Rv
14 x<>y
15 STO+ 0
16 GTO 19
17 Rv
18 x<>y
19 1
20 e^x
21 *
22 x<>y
23 GTO 04
24 RCL 0


For the Pi flavour, change steps 19 and 20 to: Pi and NOP.

The 1,000,000^th term is 278,394,443.2 and the 3,141,593^rd term is 1,601,007,657. Both in several seconds on a real 25.

The 1,234,567^th term for Pi is 9,091,632,462 and the 2,718,282^nd term is 30,446,503,22x (x being beyond the accuracy of the 25). Again, fairly quickly.

The key observation being that:
$$ \sum_{i=0}^n e^i = \frac{e^{n + 1} - 1}{e - 1} < e^{n + 1} $$
which means that the position expressed in binary determines the powers used. Note that 2¹ + 2³ + 2⁴ = 26 and 2⁰ + 2² + 2³ + 2⁴ + 2⁵ = 61 and compare to the initial examples.

Larger bases also have this property. I didn't try to prove that e is the smallest base for which this holds true, which is good because the smallest base is, unsurprisingly, two.


Pauli