On palindromic numbers, part two

In my last post on palindromic numbers, I looked at the frequencies of valid & invalid palindromic numbers and found that almost all invalid palindromes are odd. To explore why this might be, I'll now look at the differences between even and odd palindromes. (By this I mean even and odd numbers, \(n\), that have palindromic representations in a base less than \(n - 1\).)

Of evens and odds

Of the 4997 valid even palindromes through 10,000, over 62% have a length of 3 digits and just under 20% have a length of 2 digits. Palindromes of lengths ranging from 4 to 9 comprise the nearly 18% remaining. This is somewhat similar to the odds: just over 70% have length 3, a little less than 14% have length 2, and the bit more than 16% remaining have lengths ranging from 4 to 14. So the odds are spread out a bit more as far as lengths go, but they and the evens are fairly similar in that regard.

What about the frequency of bases? Do the evens have more or less of certain bases or ranges of bases? Well, as far as number of bases, the evens have a slightly higher (a little over 5.5%) diversity of (valid, lowest) bases over the first 10,000 counting numbers. I don't think that's enough to be significant, though (I'd love to be proven wrong in the comments!), so let's look at the frequencies of individual bases:

Frequencies of bases (even numbers ≤ 10,000)

Base	# of occurences
2	0
3	159
4	67
5	144
6	89
7	136
8	61
9	95
10	66
11	112
12	68
13	102
14	70
15	101
16	74
17	107
18	97
19	125
20	130
21	155
22	149
23	129
24	112
25	108
26	88
27	79
28	93
29	87
30	84
31	72
32	64
33	69
34	62
35	65
36	59
37	49
38	48
39	53
40	47
41	48
42	43
43	49
44	30
45	41
46	35
47	32
48	29
49	20
50	18
51	35
52	24
53	27
54	20
55	31
56	20
57	32
58	16
59	14
60	15
61–4986 (271 bases)	943

Frequencies of bases (odd numbers ≤ 10,000)

Base	# of occurences
2	202
3	39
4	112
5	43
6	139
7	72
8	83
9	35
10	75
11	36
12	87
13	53
14	89
15	63
16	84
17	88
18	117
19	106
20	148
21	150
22	150
23	134
24	125
25	110
26	91
27	88
28	102
29	83
30	84
31	69
32	73
33	58
34	61
35	52
36	58
37	46
38	54
39	51
40	51
41	45
42	45
43	32
44	40
45	37
46	40
47	28
48	34
49	24
50	31
51	21
52	33
53	19
54	20
55	17
56	14
57	21
58	22
59	15
60	22
61–3256 (239 bases)	860

Now here's where it gets fascinating. For the lower bases, the evens have much more odd bases and the odds have much more even bases. (Why do evens have no base-2 palindromes? A binary palindrome would have to end in a '1' — allowing leading zeroes would cause all sorts of problems — and binary numbers that end in '1' are odd.) In general, the larger the base, the lower the frequency, but around base-20, both the evens and odds see an upswing in frequencies of all bases that gradually tapers down to about base-50. It's here that we see a recurrence of the original pattern, albeit at a much smaller scale. (There is also a small uptick at base-28 for both evens and odds, and the high/low pattern makes a brief appearance in the early thirties.)

Can you take me higher?

Obviously, this particular patterning is the result of the limited number set we're working with. Does it continue for larger number groups?

Frequencies of bases (even numbers ≤ 100,000)

Base	# of occurences
2	0
3	513
4	166
5	522
6	190
7	451
8	280
9	444
10	457
11	508
12	355
13	415
14	245
15	371
16	163
17	369
18	226
19	358
20	228
21	338
22	235
23	308
24	230
25	303
26	237
27	301
28	293
29	362
30	324
31	381
32	352
33	438
34	382
35	467
36	430
37	484
38	462
39	546
40	528
41	594
42	596
43	679
44	637
45	713
46	706
47	718
48	681
49	636
50	605
51	625
52	565
53	572
54	521
55	550
56	482
57	479
58	445
59	469
60	395
61	410
62	375
63	416
64	358
65	364
66	371
67	363
68	307
69	318
70	325
71	315
72	323
73	273
74	271
75	301
76	281
77	273
78	258
79	267
80	252
81	215
82	239
83	251
84	206
85	232
86	202
87	203
88	231
89	209
90	200
91	206
92	165
93	180
94	176
95	186
96	192
97	157
98	153
99	174
100	163
101–49,990 (1831 bases)	14,306

Frequencies of bases (odd numbers ≤ 100,000)

Base	# of occurences
2	642
3	135
4	399
5	241
6	443
7	128
8	290
9	290
10	552
11	339
12	398
13	240
14	343
15	156
16	280
17	113
18	237
19	106
20	249
21	150
22	260
23	176
24	266
25	208
26	285
27	222
28	343
29	281
30	365
31	311
32	337
33	360
34	428
35	401
36	470
37	455
38	544
39	516
40	585
41	547
42	627
43	617
44	690
45	676
46	755
47	710
48	709
49	646
50	642
51	596
52	587
53	549
54	549
55	524
56	516
57	498
58	480
59	438
60	452
61	438
62	436
63	394
64	336
65	378
66	370
67	348
68	366
69	318
70	353
71	299
72	346
73	305
74	295
75	293
76	273
77	260
78	273
79	248
80	276
81	207
82	234
83	220
84	237
85	241
86	230
87	204
88	214
89	185
90	205
91	202
92	205
93	179
94	183
95	181
96	177
97	170
98	184
99	153
100	163
101–32,910 (1596 bases)	13,869

It seems so. (With one order-of-magnitude increase, at least.)

More pointedly, the early alternating pattern continues, as does the overall swelling later and the return to the alternating pattern. But— there are some hiccups in the alternating pattern now, the swelling occurs later (and is now responsible for the highest frequencies), and the later alternation is less pronounced.

I won't check higher numbers because it takes quite some time to compute these numbers (at least with the quick script I wrote for the task, and I don't wish to spend the time to see if I can make a faster program). However, if the trends I see continue, it looks like the alternating pattern becomes most significant as the upper number limit approaches infinity. (I base this on the alternations continuing mostly unhindered and the swelling moving into higher ranges. As we approach an infinite upper bound, it would follow that the swelling would approach infinity as well.) So why are even numbers likely to have their lowest-base palindrome be in an odd base and vice-versa for the odds?

I don't have an answer to that question. I'd be interested if any readers did. In the meantime, I'll look at something I noticed about one of my first lists: the invalid palindromes. For a refresher, let's look at those, this time through 10,000:

• 1
• 2
• 3
• 4
• 6
• 11
• 19
• 47
• 53
• 79
• 103
• 137
• 139
• 149
• 163
• 167
• 179
• 223
• 263
• 269
• 283
• 293
• 311
• 317
• 347
• 359
• 367
• 389
• 439
• 491
• 563
• 569
• 593
• 607
• 659
• 739
• 827
• 853
• 877
• 977
• 983
• 997
• 1019
• 1049
• 1061
• 1187
• 1213
• 1237
• 1367
• 1433
• 1439
• 1447
• 1459
• 1511
• 1553
• 1579
• 1669
• 1709
• 1753
• 1759
• 1907
• 1949
• 1993
• 1997
• 2011
• 2063
• 2087
• 2099
• 2111
• 2137
• 2179
• 2207
• 2287
• 2309
• 2339
• 2417
• 2459
• 2503
• 2657
• 2677
• 2683
• 2693
• 2713
• 2749
• 2897
• 2963
• 3023
• 3089
• 3119
• 3229
• 3253
• 3259
• 3323
• 3371
• 3407
• 3449
• 3547
• 3559
• 3583
• 3623
• 3643
• 3833
• 3847
• 4007
• 4073
• 4091
• 4099
• 4139
• 4157
• 4211
• 4283
• 4337
• 4339
• 4349
• 4391
• 4463
• 4523
• 4549
• 4643
• 4679
• 4729
• 4787
• 4871
• 4909
• 4919
• 4933
• 5011
• 5021
• 5039
• 5059
• 5099
• 5179
• 5231
• 5297
• 5303
• 5309
• 5351
• 5387
• 5417
• 5431
• 5471
• 5503
• 5527
• 5653
• 5693
• 5711
• 5791
• 5827
• 5839
• 5939
• 6047
• 6067
• 6079
• 6089
• 6131
• 6199
• 6229
• 6247
• 6269
• 6277
• 6311
• 6343
• 6359
• 6389
• 6551
• 6599
• 6653
• 6793
• 6871
• 6947
• 6983
• 6991
• 7019
• 7079
• 7159
• 7213
• 7247
• 7283
• 7433
• 7487
• 7691
• 7817
• 7877
• 7949
• 7963
• 8017
• 8069
• 8089
• 8123
• 8147
• 8221
• 8243
• 8287
• 8291
• 8293
• 8423
• 8539
• 8573
• 8669
• 8699
• 8783
• 8863
• 8941
• 9043
• 9059
• 9067
• 9173
• 9209
• 9227
• 9277
• 9337
• 9341
• 9377
• 9419
• 9421
• 9533
• 9587
• 9643
• 9689
• 9739
• 9781
• 9887

I've highlighted the non-prime numbers. There are only three: 1, 4 and 6. Aha! So, not only are all invalid palindromes above 6 odd, they are all prime as well! (Assuming this continues beyond 10,000. I have no proof it does, but so far it works.)

Roadblocks

Unfortunately, I have no idea where to go with this information. I can't see any patterns in the primes that do not have valid palindromes vs. those that do. Moreover, I have no clue of the mechanics behind a number having a palindrome in a particular base over another. We have some clues, but no full set of numbers that do or do not have valid palindromes. Hmm….