Strictly non-palindromic number: Difference between revisions

Revision as of 19:55, 8 January 2006

Revision as of 16:56, 10 January 2006

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To test whether a number ''n'' foris strictstrictly non-palindromicitypalindromic, it must be verified that ''n'' is non-palindromic in all bases up to ''n'' − 2. ThatThe upperreasons limitfor maythese seemupper arbitrary, but actually makeslimit senseare:

==About the definition==

To test a number ''n'' for strict non-palindromicity, it must be verified that ''n'' is non-palindromic in all bases up to ''n'' − 2. That upper limit may seem arbitrary, but actually makes sense:

*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'';

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The reader can easily verify that in each case (1) the base ''b'' is in the range 2&nbsp;&le;&nbsp;''b''&nbsp;&le;&nbsp;''n''&nbsp;&minus;&nbsp;2, and (2) the digits ''a''_''i'' of each palindrome are in the range 0&nbsp;&le;&nbsp;''a''_''i''&nbsp;<&nbsp;''b'', given that ''n''&nbsp;>&nbsp;6. These conditions may fail if ''n''&nbsp;&le;&nbsp;6, which explains why the non-prime numbers 1, 4 and 6 are strictly non-palindromic nevertheless.

This concludes the proof thatTherefore, all strictly non-palindromic ''n''&nbsp;>&nbsp;6 are prime.

==References==