Strictly non-palindromic number

Strictly non-palindromic number

A strictly non-palindromic number is an integer "n" that is not palindromic in any numeral system with a base "b" in the range 2 ≤ "b" ≤ "n" − 2. For example, the number six is written as 110 in base 2, 20 in base 3 and 12 in base 4, none of which is a palindrome—so 6 is strictly non-palindromic.

The sequence of strictly non-palindromic numbers OEIS|id=A016038 starts:

:1, 2, 3, 4, 6, 11, 19, 47, 53, 79, 103, 137, 139, 149, 163, 167, 179, 223, 263, 269, 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. 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, then "n" is written 22 (a palindrome) in base "b" = "n"/2 − 1.Otherwise "n" is odd. Write "n" = "p"&thinsp;&middot;&thinsp;"m", where "p" is the smallest odd prime factor of "n". Then clearly "p" &le; "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 &le; "b" &le; "n" − 2, and (2) the digits "a""i" of each palindrome are in the range 0 &le; "a""i" < "b", given that "n" > 6. These conditions may fail if "n" &le; 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.

References

* Sequence from the On-Line Encyclopedia of Integer Sequences


Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • 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… …   Wikipedia

  • 300 (number) — This article is about the numbers 300 to 399. For other uses of 300, see 300 (disambiguation). For the guitar, see Gibson ES 335. For the British tilting train, see British Rail Class 390. For the Dada magazine, see 391 (magazine). For the… …   Wikipedia

  • 900 (number) — For the year 900, see 900 BC or 900 AD. 900 (nine hundred) is the natural number following 899 and preceding 901. It is the square of 30 and the sum of Euler s totient function for the first 54 integers. In base 10 it is a Harshad number. List of …   Wikipedia

  • 500 (number) — For other uses, see 500 (disambiguation). ← 499 501 → 500 List of numbers Integers …   Wikipedia

  • 800 (number) — This article is about the number 800. For the Common Era Year 800, see 800. For other uses, see 800 (disambiguation) 800 (eight hundred) is the natural number following 799 and preceding 801. List of numbers Integers ← 0 100 200 300 400 500 600… …   Wikipedia

  • 600 (number) — For the year 600, see 600 BC or 600 AD. ← 599 601 → 600 List of numbers Integers ← …   Wikipedia

  • 400 (number) — ← 399 401 → 400 List of numbers Integers ← 0 100 200 …   Wikipedia

  • 700 (number) — This article is about the numbers 700 through 799; for each individual number, see its section below. 700 (seven hundred) is the natural number following 699 and preceding 701. List of numbers Integers ← 0 100 200 300 400 500 600 700 800 …   Wikipedia

  • 260 (number) — 260 (two hundred [and] sixty) is the magic constant of the n times; n normal magic square and n Queens Problem for n = 8, the size of an actual chess board.260 is also the magic constant of the Franklin magic square devised by Benjamin… …   Wikipedia

  • 311 (number) — 311 is the natural number after 310 and before 312.In mathematics311 is a twin prime with 313; an Eisenstein prime with no imaginary part and real part of the form 3n 1; a Gaussian prime with no imaginary part and real part of the form 4n 1; and… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”