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….

On palindromic numbers

Much of this is taken from my comment to a post titled Can every number be written as a palindrome in some base? on MathFour.com. The original post asks whether every number has a palindromic representation in some base.

What's a palindrome?

First, some definitions, for those who don't read the original post. A palindrome is read the same backwards and forwards. E.g.: racecar or A man, a plan, a canal: Panama. Note that only the letters are considered, not spaces or punctuation. A palindromic number has the same properties: 121 and 98,766,789 are both palindromes.

We commonly use base-10 numbers, but computers use base-2 (binary) and we encounter base-16 (hexadecimal) in computing, often to represent colors. The number of a base determines how many symbols are possible for each digit. (0-9 for base-10, 0 & 1 for base-2, 0-9 plus A-F for base-16, etc.) More to the point, it determines the value of each number-place. The first place is how many singles, the next is how many of the base value, and subsequent places are how many of increasing powers of the base value. In our everyday decimal, or base-10, numbers, this would be ones, tens (the base), hundreds (\(10^2\)), thousands (\(10^3\)), etc.

The original post asked whether all numbers could be written as a palindrome in some base, whether it be base-10 (121 for example) or another base. As it figures, every number can be written as a palindrome in the base one less than the number (\(n - 1\)) such as 3 being 11 in base-2. This is due to the definition of bases, where the first '1' is the base value (here, one less than the number) and the second '1' is the value one. In other words, \((n - 1) + 1 = n\). [Also note that in bases > \(n\) the number will be a single digit. This, too, is a palindrome, albeit a lame one.]

A limited limit

So, what's a useful limiting factor? The original post looks at those numbers which have a palindrome with a base ≤ 10; i.e., one that would be written using only the digits 0–9. For the sake of completeness, let's look at those numbers that are invalid based on that definition (the original post included too many):

Base > 10 (1–100)

Number	Base	Palindrome
19	18	[1, 1]
39	12	[3, 3]
47	46	[1, 1]
53	52	[1, 1]
58	28	[2, 2]
69	22	[3, 3]
75	14	[5, 5]
76	18	[4, 4]
79	78	[1, 1]
84	11	[7, 7]
87	28	[3, 3]
90	14	[6, 6]
94	46	[2, 2]
95	18	[5, 5]
96	11	[8, 8]

Note: The palindromes are shown as a bracketed list with the value of each place value. This is to make larger bases easily represented and readable without having to remember what 'R' means, for example (it's 27), and without having to invent single-character representations of values larger than 35.

I find this to be too arbitrary. Yes, it's easier to look & think about for us humans who're used to base-10 numbers, but where's the fun there? (Also note that none of those invalid palindromes use digits > 9, highlighting the arbitrariness of the imposed limit.)

I tend more towards general solutions to problems, so I think a more interesting question is: What is the lowest base in which \(n\) is a palindrome? (Obviously, we're omitting base-1 numbers as they are all palindromes.)

Also, since all numbers are palindromes in bases greater than \(n - 1\) (excluding base-\(n\)), why not limit "valid" palindromes to those less than \(n - 1\)? So the question becomes: Does \(n\) have a palindrome in a base less than base-\((n - 1)\)? Is there a pattern to these numbers? Do they occur more, less or equally frequent as we look at larger sets of integers?

Python to the rescue!

Of course, computers make these things so much easier to look at. I made a quick Python script to examine palindromic numbers. The benefits here are I don't have to worry about inventing a character for each digit. I can just leave the decimal value of that digit as-is and place the digits in a list. So, here are some results for integers 1–100:

The full list (1–100)

Number	Base	Palindrome
1	2	[1]
2	3	[2]
3	2	[1, 1]
4	3	[1, 1]
5	2	[1, 0, 1]
6	5	[1, 1]
7	2	[1, 1, 1]
8	3	[2, 2]
9	2	[1, 0, 0, 1]
10	3	[1, 0, 1]
11	10	[1, 1]
12	5	[2, 2]
13	3	[1, 1, 1]
14	6	[2, 2]
15	2	[1, 1, 1, 1]
16	3	[1, 2, 1]
17	2	[1, 0, 0, 0, 1]
18	5	[3, 3]
19	18	[1, 1]
20	3	[2, 0, 2]
21	2	[1, 0, 1, 0, 1]
22	10	[2, 2]
23	3	[2, 1, 2]
24	5	[4, 4]
25	4	[1, 2, 1]
26	3	[2, 2, 2]
27	2	[1, 1, 0, 1, 1]
28	3	[1, 0, 0, 1]
29	4	[1, 3, 1]
30	9	[3, 3]
31	2	[1, 1, 1, 1, 1]
32	7	[4, 4]
33	2	[1, 0, 0, 0, 0, 1]
34	4	[2, 0, 2]
35	6	[5, 5]
36	5	[1, 2, 1]
37	6	[1, 0, 1]
38	4	[2, 1, 2]
39	12	[3, 3]
40	3	[1, 1, 1, 1]
41	5	[1, 3, 1]
42	4	[2, 2, 2]
43	6	[1, 1, 1]
44	10	[4, 4]
45	2	[1, 0, 1, 1, 0, 1]
46	4	[2, 3, 2]
47	46	[1, 1]
48	7	[6, 6]
49	6	[1, 2, 1]
50	7	[1, 0, 1]
51	2	[1, 1, 0, 0, 1, 1]
52	3	[1, 2, 2, 1]
53	52	[1, 1]
54	8	[6, 6]
55	4	[3, 1, 3]
56	3	[2, 0, 0, 2]
57	5	[2, 1, 2]
58	28	[2, 2]
59	4	[3, 2, 3]
60	9	[6, 6]
61	6	[1, 4, 1]
62	5	[2, 2, 2]
63	2	[1, 1, 1, 1, 1, 1]
64	7	[1, 2, 1]
65	2	[1, 0, 0, 0, 0, 0, 1]
66	10	[6, 6]
67	5	[2, 3, 2]
68	3	[2, 1, 1, 2]
69	22	[3, 3]
70	9	[7, 7]
71	7	[1, 3, 1]
72	5	[2, 4, 2]
73	2	[1, 0, 0, 1, 0, 0, 1]
74	6	[2, 0, 2]
75	14	[5, 5]
76	18	[4, 4]
77	10	[7, 7]
78	5	[3, 0, 3]
79	78	[1, 1]
80	3	[2, 2, 2, 2]
81	8	[1, 2, 1]
82	3	[1, 0, 0, 0, 1]
83	5	[3, 1, 3]
84	11	[7, 7]
85	2	[1, 0, 1, 0, 1, 0, 1]
86	6	[2, 2, 2]
87	28	[3, 3]
88	5	[3, 2, 3]
89	8	[1, 3, 1]
90	14	[6, 6]
91	3	[1, 0, 1, 0, 1]
92	6	[2, 3, 2]
93	2	[1, 0, 1, 1, 1, 0, 1]
94	46	[2, 2]
95	18	[5, 5]
96	11	[8, 8]
97	8	[1, 4, 1]
98	5	[3, 4, 3]
99	2	[1, 1, 0, 0, 0, 1, 1]
100	3	[1, 0, 2, 0, 1]

Now, "invalid" palindromes (note how few there are):

Base ≥ \((n - 1)\) (1–100)

Number	Base	Palindrome
1	2	[1]
2	3	[2]
3	2	[1, 1]
4	3	[1, 1]
6	5	[1, 1]
11	10	[1, 1]
19	18	[1, 1]
47	46	[1, 1]
53	52	[1, 1]
79	78	[1, 1]

Only 10% of numbers through 100 have only the default palindrome. Interestingly, none of these numbers use a digit greater than 8 in their palindromic representation. In fact … the first number to use 10 (or greater) in their lowest-base palindromic representation is 120 (base 11, [10, 10]). (108 is the first to use 9.)

What's the frequency?

How about the frequencies of these invalid palindromes? Here's a look at the percentages for each group of 100 numbers through 10,000:

Frequencies of invalid palindromes

Range	%
1–100	10
101–200	7
201–300	5
301–400	6
401–500	2
501–600	3
601–700	2
701–800	1
801–900	3
901–1000	3
1001–1100	3
1101–1200	1
1201–1300	2
1301–1400	1
1401–1500	4
1501–1600	3
1601–1700	1
1701–1800	3
1801–1900	0
1901–2000	4
2001–2100	4
2101–2200	3
2201–2300	2
2301–2400	2
2401–2500	2
2501–2600	1
2601–2700	4
2701–2800	2
2801–2900	1
2901–3000	1
3001–3100	2
3101–3200	1
3201–3300	3
3301–3400	2
3401–3500	2
3501–3600	3
3601–3700	2
3701–3800	0
3801–3900	2
3901–4000	0
4001–4100	4
4101–4200	2
4201–4300	2
4301–4400	4
4401–4500	1
4501–4600	2
4601–4700	2
4701–4800	2
4801–4900	1
4901–5000	3
5001–5100	5
5101–5200	1
5201–5300	2
5301–5400	4
5401–5500	3
5501–5600	2
5601–5700	2
5701–5800	2
5801–5900	2
5901–6000	1
6001–6100	4
6101–6200	2
6201–6300	4
6301–6400	4
6401–6500	0
6501–6600	2
6601–6700	1
6701–6800	1
6801–6900	1
6901–7000	3
7001–7100	2
7101–7200	1
7201–7300	3
7301–7400	0
7401–7500	2
7501–7600	0
7601–7700	1
7701–7800	0
7801–7900	2
7901–8000	2
8001–8100	3
8101–8200	2
8201–8300	5
8301–8400	0
8401–8500	1
8501–8600	2
8601–8700	2
8701–8800	1
8801–8900	1
8901–9000	1
9001–9100	3
9101–9200	1
9201–9300	3
9301–9400	3
9401–9500	2
9501–9600	2
9601–9700	2
9701–9800	2
9801–9900	1
9901–10000	0

There are very few numbers without a valid palindromic representation. In fact, through 10,000 there are only 2.22 per hundred. While there are only 9 instances of no valid palindromes in the arbitrary 100-number blocks above, there are in fact 22 different stretches of triple-digit runs without an invalid palindrome, the longest of these being 203! This includes the 112-number run that finishes with 10,000 and possibly goes longer. Excluding this final, partial run, the mean length of a run is a little over 44 ⅓.

Runs of valid palindromes

Invalid number	Valid numbers preceding
1	0
2	0
3	0
4	0
6	1
11	4
19	7
47	27
53	5
79	25
103	23
137	33
139	1
149	9
163	13
167	3
179	11
223	43
263	39
269	5
283	13
293	9
311	17
317	5
347	29
359	11
367	7
389	21
439	49
491	51
563	71
569	5
593	23
607	13
659	51
739	79
827	87
853	25
877	23
977	99
983	5
997	13
1019	21
1049	29
1061	11
1187	125
1213	25
1237	23
1367	129
1433	65
1439	5
1447	7
1459	11
1511	51
1553	41
1579	25
1669	89
1709	39
1753	43
1759	5
1907	147
1949	41
1993	43
1997	3
2011	13
2063	51
2087	23
2099	11
2111	11
2137	25
2179	41
2207	27
2287	79
2309	21
2339	29
2417	77
2459	41
2503	43
2657	153
2677	19
2683	5
2693	9
2713	19
2749	35
2897	147
2963	65
3023	59
3089	65
3119	29
3229	109
3253	23
3259	5
3323	63
3371	47
3407	35
3449	41
3547	97
3559	11
3583	23
3623	39
3643	19
3833	189
3847	13
4007	159
4073	65
4091	17
4099	7
4139	39
4157	17
4211	53
4283	71
4337	53
4339	1
4349	9
4391	41
4463	71
4523	59
4549	25
4643	93
4679	35
4729	49
4787	57
4871	83
4909	37
4919	9
4933	13
5011	77
5021	9
5039	17
5059	19
5099	39
5179	79
5231	51
5297	65
5303	5
5309	5
5351	41
5387	35
5417	29
5431	13
5471	39
5503	31
5527	23
5653	125
5693	39
5711	17
5791	79
5827	35
5839	11
5939	99
6047	107
6067	19
6079	11
6089	9
6131	41
6199	67
6229	29
6247	17
6269	21
6277	7
6311	33
6343	31
6359	15
6389	29
6551	161
6599	47
6653	53
6793	139
6871	77
6947	75
6983	35
6991	7
7019	27
7079	59
7159	79
7213	53
7247	33
7283	35
7433	149
7487	53
7691	203
7817	125
7877	59
7949	71
7963	13
8017	53
8069	51
8089	19
8123	33
8147	23
8221	73
8243	21
8287	43
8291	3
8293	1
8423	129
8539	115
8573	33
8669	95
8699	29
8783	83
8863	79
8941	77
9043	101
9059	15
9067	7
9173	105
9209	35
9227	17
9277	49
9337	59
9341	3
9377	35
9419	41
9421	1
9533	111
9587	53
9643	55
9689	45
9739	49
9781	41
9887	105

Looking at these two frequency graphs, it seems invalid palindromes, while a little more frequent in lower integers, occur otherwise without a discernible pattern.

Where do we go from here?

While we're having fun, let's look at some specific groups of numbers. Of the 222 invalid palindromes from 1–10,000, a staggering 219 are odd, with only 3 even. These are 2, 4 & 6 — the first three even numbers. Now we're getting somewhere — almost all invalid palindromes are odd! So why do almost all evens (assuming there continue to be none beyond 5,000,276 — the highest I've figured out so far through a brute force program running while I type this post — which is not certain) have valid palindromes? Maybe we'll get some insight if we examine the even palindromes. Join me in the next post for just that.