Editing Palindromic number (section)

==Decimal palindromic numbers==
All numbers with one digit are palindromic, so in [[Decimal|base 10]] there are ten palindromic numbers with one digit:
:{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

There are 9 palindromic numbers with two digits:
:{11, 22, 33, 44, 55, 66, 77, 88, 99}.

All palindromic numbers with an even number of digits are divisible by [[11 (number)|11]].<ref>{{cite web |title=The Prime Glossary: palindromic prime |url=https://t5k.org/glossary/page.php?sort=PalindromicPrime |website=[[PrimePages]] |access-date=11 July 2023}}</ref>

There are 90 palindromic numbers with three digits (Using the [[rule of product]]: 9 choices for the first digit - which determines the third digit as well - multiplied by 10 choices for the second digit):
:{101, 111, 121, 131, 141, 151, 161, 171, 181, 191, ..., 909, 919, 929, 939, 949, 959, 969, 979, 989, 999}

There are likewise 90 palindromic numbers with four digits (again, 9 choices for the first digit multiplied by ten choices for the second digit. The other two digits are determined by the choice of the first two):
:{1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, ..., 9009, 9119, 9229, 9339, 9449, 9559, 9669, 9779, 9889, 9999},
so there are 199 palindromic numbers smaller than 10<sup>4</sup>.

There are 1099 palindromic numbers smaller than 10<sup>5</sup> and for other exponents of 10<sup>n</sup> we have: 1999, 10999, 19999, 109999, 199999, 1099999, ... {{OEIS|id=A070199}}. The number of palindromic numbers which have some other property are listed below:
{| class="wikitable"
|-
! &nbsp;
! 10<sup>1</sup>
! 10<sup>2</sup>
! 10<sup>3</sup>
! 10<sup>4</sup>
! 10<sup>5</sup>
! 10<sup>6</sup>
! 10<sup>7</sup>
! 10<sup>8</sup>
! 10<sup>9</sup>
! 10<sup>10</sup>
|-
! style="font-weight:normal; text-align:left" | ''n'' [[Natural number|natural]]
| 10
| 19
| 109
| 199
| 1099
| 1999
| 10999
| 19999
| 109999
| 199999
|-
! style="font-weight:normal; text-align:left" | ''n'' [[even and odd numbers|even]]
| 5
| 9
| 49
| 89
| 489
| 889
| 4889
| 8889
| 48889
| 88889
|-
! style="font-weight:normal; text-align:left" | ''n'' [[odd number|odd]]
| 5
| 10
| 60
| 110
| 610
| 1110
| 6110
| 11110
| 61110
| 111110
|-
! style="font-weight:normal; text-align:left" | ''n'' [[square number|square]]
| colspan="2" | 4
| colspan="2" | 7
| 14
| 15
| colspan="2" | 20
| colspan="2" | 31
|-
! style="font-weight:normal; text-align:left" | ''n'' [[Cube (algebra)|cube]]
| colspan="2" | 3
| 4
| colspan="3" | 5
| colspan="3" | 7
| 8
|-
! style="font-weight:normal; text-align:left" | ''n'' [[prime number|prime]]
| 4
| 5
| colspan="2" | 20
| colspan="2" | 113
| colspan="2" | 781
| colspan="2" | 5953
|-
! style="font-weight:normal; text-align:left" | ''n'' [[square-free integer|squarefree]]
| 6
| 12
| 67
| 120
| 675
| 1200
| 6821
| 12160
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' non-squarefree ([[Möbius function|μ(''n'')]]=0)
| 4
| 7
| 42
| 79
| 424
| 799
| 4178
| 7839
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' square with prime root<ref>{{OEIS|A065379}} The next example is 19 digits - 900075181570009.</ref>
| colspan="1" | 2
| colspan="2" | 3
| colspan="6" | 5
|-
! style="font-weight:normal; text-align:left" | ''n'' with an even number of distinct [[prime factor]]s (μ(''n'')=1)
| 2
| 6
| 35
| 56
| 324
| 583
| 3383
| 6093
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' with an odd number of distinct prime factors (μ(''n'')=-1)
| 4
| 6
| 32
| 64
| 351
| 617
| 3438
| 6067
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' even with an odd number of prime factors
| 1
| 2
| 9
| 21
| 100
| 180
| 1010
| 6067
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' even with an odd number of distinct prime factors
| 3
| 4
| 21
| 49
| 268
| 482
| 2486
| 4452
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' odd with an odd number of prime factors
| 3
| 4
| 23
| 43
| 251
| 437
| 2428
| 4315
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' odd with an odd number of distinct prime factors
| 4
| 5
| 28
| 56
| 317
| 566
| 3070
| 5607
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' even squarefree with an even number of (distinct) prime factors
| 1
| 2
| 11
| 15
| 98
| 171
| 991
| 1782
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' odd squarefree with an even number of (distinct) prime factors
| 1
| 4
| 24
| 41
| 226
| 412
| 2392
| 4221
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' odd with exactly 2 prime factors
| 1
| 4
| 25
| 39
| 205
| 303
| 1768
| 2403
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' even with exactly 2 prime factors
| 2
| 3
| colspan="2" | 11
| colspan="2" | 64
| colspan="2" | 413
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' even with exactly 3 prime factors
| 1
| 3
| 14
| 24
| 122
| 179
| 1056
| 1400
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' even with exactly 3 distinct prime factors
| 0
| 1
| 18
| 44
| 250
| 390
| 2001
| 2814
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' odd with exactly 3 prime factors
| 0
| 1
| 12
| 34
| 173
| 348
| 1762
| 3292
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' [[Carmichael number]]
| 0
| 0
| 0
| 0
| 0
| 1
| 1
| 1
| 1
| 1
|-
! style="font-weight:normal; text-align:left" | ''n'' for which [[Divisor function|σ(''n'')]] is palindromic
| 6
| 10
| 47
| 114
| 688
| 1417
| 5683
| +
| +
| +
|}

===Perfect powers===
There are many palindromic [[perfect power]]s ''n''<sup>''k''</sup>, where ''n'' is a natural number and ''k'' is 2, 3 or 4.
* Palindromic [[Square number|squares]]: 0, 1, 4, 9, 121, 484, 676, 10201, 12321, 14641, 40804, 44944, ... {{OEIS|id=A002779}}
* Palindromic [[Cube (algebra)|cubes]]: 0, 1, 8, 343, 1331, 1030301, 1367631, 1003003001, ... {{OEIS|id=A002781}}
* Palindromic [[fourth power]]s: 0, 1, 14641, 104060401, 1004006004001, ... {{OEIS|id=A186080}}

The first nine terms of the sequence 1<sup>2</sup>, 11<sup>2</sup>, 111<sup>2</sup>, 1111<sup>2</sup>, ... form the palindromes 1, 121, 12321, 1234321, ... {{OEIS|id=A002477}}

The only known non-palindromic number whose cube is a palindrome is 2201, and it is a conjecture the fourth root of all the palindrome fourth powers are a palindrome with 100000...000001 (10<sup>n</sup> + 1).

[[Gustavus Simmons]] conjectured there are no palindromes of form ''n''<sup>''k''</sup> for ''k'' > 4 (and ''n'' > 1).<ref>Murray S. Klamkin (1990), ''Problems in applied mathematics: selections from SIAM review'', [https://books.google.com/books?id=WI9ZGl3M8bYC&pg=PA520 p. 520].</ref>