A strictly non-palindromic number is an integer n that is not palindromic

Palindromic number

A palindromic number or numeral palindrome is a 'symmetrical' number like 16461, that remains the same when its digits are reversed. The term palindromic is derived from palindrome, which refers to a word like rotor that remains unchanged under reversal of its letters...

 in any numeral system

Numeral system

A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....

 with a base

Radix

In mathematical numeral systems, the base or radix for the simplest case is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.In any numeral...

 b in the range 2 ≤ b ≤ n − 2. For example, the number six is written as 110 in base 2

Binary numeral system

The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...

, 20 in base 3

Ternary numeral system

Ternary is the base- numeral system. Analogous to a bit, a ternary digit is a trit . One trit contains \log_2 3 bits of information...

 and 12 in base 4

Quaternary numeral system

Quaternary is the base- numeral system. It uses the digits 0, 1, 2 and 3 to represent any real number.It shares with all fixed-radix numeral systems many properties, such as the ability to represent any real number with a canonical representation and the characteristics of the representations of...

, none of which is a palindrome—so 6 is strictly non-palindromic.

The sequence of strictly non-palindromic numbers starts:

1, 2, 3, 4, 6, 11

11 (number)

11 is the natural number following 10 and preceding 12.Eleven is the first number which cannot be counted with a human's eight fingers and two thumbs additively. In English, it is the smallest positive integer requiring three syllables and the largest prime number with a single-morpheme name...

, 19

19 (number)

19 is the natural number following 18 and preceding 20. It is a prime number.In English speech, the numbers 19 and 90 are often confused. When carefully enunciated, they differ in which syllable is stressed: 19 vs 90...

, 47

47 (number)

47 is the natural number following 46 and preceding 48.-In mathematics:Forty-seven is the fifteenth prime number, a safe prime, the thirteenth supersingular prime, and the sixth Lucas prime. Forty-seven is a highly cototient number...

, 53

53 (number)

53 is the natural number following 52 and preceding 54.-In mathematics:Fifty-three is the 16th prime number. It is also an Eisenstein prime....

, 79

79 (number)

Seventy-nine is the natural number following 78 and preceding 80.79 may represent:-In mathematics:*An odd number*The smallest number that can't be represented as a sum of fewer than 19 fourth powers*A strictly non-palindromic number...

, 103

103 (number)

103 is the natural number following 102 and preceding 104.-In mathematics:One hundred [and] three is the 27th prime number. The previous prime is 101, making them both twin primes...

, 137

137 (number)

137 is the natural number following 136 and preceding 138.-In mathematics :One hundred [and] thirty-seven is the 33rd prime number; the next is 139, with which it comprises a twin prime, and thus 137 is a Chen prime. 137 is an Eisenstein prime with no imaginary part and a real part of the form 3n -...

, 139

139 (number)

139 is the natural number following 138 and preceding 140.-In mathematics:One hundred [and] thirty-nine is the 34th prime number, so it is divisible only by itself and 1. It is a twin prime with 137. Because 141 is a semiprime, 139 is a Chen prime...

, 149

149 (number)

149 is the natural number between 148 and 150. It is also a prime number.-In mathematics:*149 is the 35th prime number, and with the next prime number, 151, is a twin prime, thus 149 is a Chen prime. 149 is a strong prime in the sense that it is more than the arithmetic mean of its two neighboring...

, 163

163 (number)

163 is the natural number following 162 and preceding 164.-In mathematics:163 is a strong prime in the sense that it is greater than the arithmetic mean of its two neighboring primes...

, 167

167 (number)

167 is the natural number following 166 and preceding 168.-In mathematics:* 167 is an odd number* 167 is a Chen prime, since the next odd number, 169, is a square of a prime...

, 179

179 (number)

179 is the natural number following 178 and preceding 180.-In mathematics:* 179 is an odd number* 179 is a deficient number, as 1 is less than 179* 179 is a Gaussian number* 179 is an odious number* 179 is a square-free number...

, 223

223 (number)

223 is the natural number between 222 and 224. It is also a prime number.-In mathematics:223 is a long prime, a strong prime, a lucky prime and a sexy prime .223 is the fourth Carol number and the third to be prime....

, 263

263 (number)

263 is the natural number between 262 and 264. It is also a prime number.-In mathematics:263 is an irregular prime, an Eisenstein prime, a long prime, a Chen prime, a Gaussian prime, a happy prime, a sexy prime, a safe prime, and a Higgs prime....

, 269

269 (number)

269 is the natural number between 268 and 270. It is also a prime number.-In mathematics:269 is a regular prime, an Eisenstein prime with no imaginary part, a long prime, a Chen prime, a Pillai prime, a Pythagorean prime, a twin prime, a sexy prime, a Higgs prime, a strong prime, and a highly...

, 283, 293, …

To test whether a number n is strictly non-palindromic, it must be verified that n is non-palindromic in all bases up to n − 2. The reasons for this upper limit are:

• any n ≥ 3 is written 11 in base n − 1, so n is palindromic in base n − 1;
• any n ≥ 2 is written 10 in base n, so any n is non-palindromic in base n;
• any n ≥ 1 is a single-digit number in any base b > n, so any n is palindromic in all such bases.

Thus it can be seen that the upper limit of n − 2 is necessary to obtain a mathematically 'interesting' definition.

For n < 4 the range of bases is empty, so these numbers are strictly non-palindromic in a trivial way.

Properties

All strictly non-palindromic numbers beyond 6 are prime

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

. To see why composite n > 6 cannot be strictly non-palindromic, for each such n a base b must be shown to exist where n is palindromic.

• If n is even
  Even and odd numbers
  In mathematics, the parity of an object states whether it is even or odd.This concept begins with integers. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2...
  
  , then n is written 22 (a palindrome) in base b = n/2 − 1.

Otherwise n is odd. Write n = p · m, where p is the smallest odd prime factor of n. Then clearly p ≤ m.

• If p = m = 3, then n = 9 is written 1001 (a palindrome) in base b = 2.
• If p = m > 3, then n is written 121 (a palindrome) in base b = p − 1.

Otherwise p < m − 1. The case p = m − 1 cannot occur because both p and m are odd.

• Then n is written pp (the two-digit number with each digit equal to p, a palindrome) in base b = m − 1.

The reader can easily verify that in each case (1) the base b is in the range 2 ≤ b ≤ n − 2, and (2) the digits a_i of each palindrome are in the range 0 ≤ a_i < b, given that n > 6. These conditions may fail if n ≤ 6, which explains why the non-prime numbers 1, 4 and 6 are strictly non-palindromic nevertheless.

Therefore, all strictly non-palindromic n > 6 are prime.