There are many other similar infinite sequences. This works in any base which allows the "digit" 2, that is any base bigger than 2. Positions may be of any length up to hundreds of digits but larger numbers may slow down some calculations.All numbers N are palindromic in every base beyond N+1 as well as in base 1 and base N−1 as we saw above.We can regard these as the 'obvious' (mathematicians use the term 'trivial') cases and other answers are the 'interesting' ones when looking forbases in which a number is palindromic.įor example, 11 2 = 121, 101 2 = 10201, 1001 2 = 1002001, 10001 2= 100020001. The position of a palindrome is its index number in the list of that base's palindromes where the first palindromeat position 1 is always 0. Palindromic BASE Numbers Calculator Bases in this Calculator may be any positive number greater than 1 with a size limit of about 15 digits or Factorial base: ! or Fibonacci base: F. Use the Calculator here to try out some numbers and bases.Ī set is a collection of distinct numbers, shown in increasing order.Ī list is a collection of numbers which may occur more than once, shown in order. All other numbers in all other bases need only 3 at most. The reason base 2 is an exception is that palindromes in base 2 begin and end with 1.A sum of 3 binary palindromes will therefore end in 1 and so must be odd.Any even number will need an even number of palindromes in its sum. Then in June 2016 Rajasekaran, Shallit and Smith of the University of Waterloo filled in the gap for bases 2, 3 and 4and proved that we can add bases 3 and 4 to the list but sometimes four base 2 palindromes are needed but a maximum of fourapplies to all binary numbers. They proved it applies to all bases from 5 upwards. Recently (2016) Cilleruelo, Luca and Baxter proved that 10 is not unique in having the property that every number is the sum of 3 palindromes in that base. A261132and since this is always possible, we know that all entries in this sequence are at least 1. How many can you think of? Show someĮvery integer is a sum of at most 3 palindromes in base 10. It is perhaps surprising at first to learn that all numbers are palindromes in some base. On this page we will not count such numbers as palindromes. We usually don't write initial 0s in a number but if we did then 10 would also be a palindrome since it is 010. Whether a number is a palindrome or not depends on which base it is represented in but all numbersare palindromic in some base.Īll numbers are palindromes in some baseFor example 21 is not a palindromein base 10, but it is in base 2: 21 = 10101 2. We can have palindromic numbers also for example 19291. Perhaps your name is a palindrome (BOB, ANNA, EVE)? PalindromesEvery family has a palindrome in it: MUM and DAD, since a palindrome is a word that is the same when reversed. section of questions to start your own investigations.The calculator iconindicates that there is a live interactive calculator in that section. Contents of this page The icon means there is a You Do the Maths.
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