On
The Big Bang Theory's episode "The Athenaeum Allocation", Sheldon mentions to Amy that for their proposed wedding date, May 12,
"The month squared equals the square of the sum of the members of the set of the prime factors of the day. Isn't that romantic?"
Amy's reply contorts a Shakespeare quote: "shall I compare thee to a day that's also a really weird math problem?"
It's indeed weird as the squaring is introduced for no reason at all.
But also, how rare is the circumstance? What other dates of the year exhibit the property? Remember that because of the set-membership criterion, each prime factor is included in the sum only once: 12 has prime factors 2 and 3, and those add up to 5, for May.
There are quite a few good candidates.
Let D = day, M = month, PF_i = i-th prime factor of D (i = 1, ..., n)
Then, M = sigma {i=1,... n} PF_i
D PF M Good Date
2 2 2 Feb 2
3 3 3 Mar 3
4 2 2 Feb 4
5 5 5 May 5
6 2, 3 5 May 6
7 7 7 Jul 7
8 2 2 Feb 8
9 3 3 Mar 9
10 2, 5 7 Jul 10
11 11 11 Nov 11
12 2, 3 5 May 12
13 13 (inadmissible)
14 2, 7 9 Sep 14
15 3, 5 8 Aug 18
16 2 2 Feb 16
17 17 (inadmissible)
18 2, 3 5 May 18
19 19 (inadmissible)
20 2, 5 7 Jul 20
21 3, 7 10 Oct 21
22 2, 11 (inadmissible)
23 23 (inadmissible)
24 2, 3 5 May 24
25 5 5 May 25
26 2, 13 (inadmissible)
27 3 3 Mar 27
28 2, 7 9 Sep 28
29 29 (inadmissible)
30 2, 3, 5 10 Oct 10
31 31 (inadmissible)
In summary: Jan (none except the trivial Jan 1)
Feb 2, 4, 8, 16
Mar 3, 9, 27
Apr (none)
May 5, 6, 12, 18, 24, 25
Jun (none)
Jul 7, 10, 20
Aug 8
Sep 14, 28
Oct 10, 21
Nov 11
Dec (none)
Thanks BBT and Charlie!
Spoiler (I think . . .)
