Talk:Automorphic number

Other Bases?

other bases? Revolver 09:28, 10 Nov 2004 (UTC)

Merging Circular number to here

As it has been concluded on the talk page for Circular number that this is a synonym of "Automorphic number", I think that they should be merged, after which Circular number should be made a redirect page to Automorphic number. Or the other way around, but since the name in OEIS is "automorphic number", this seems the best to me. --Lambiam^Talk 02:03, 6 June 2006 (UTC)

Number is wrong

The 1000 digit number is wrong for k > 975. Just check it in any program which makes exact calculations. —Preceding unsigned comment added by 79.170.121.81 (talk) 06:13, 11 October 2009 (UTC)

Has now been fixed, I think. Gandalf61 (talk) 07:18, 11 October 2009 (UTC)

You are right —Preceding unsigned comment added by 79.170.121.81 (talk) 07:20, 11 October 2009 (UTC)

Linear time calculation algorithm

"Algorithm exists to generate automorphic number with linear complexity O(n) if n-1 number is known[1]"

This statement is true. Just check this program !

9 isn't prime

Currently under "Other radixes" it says "A prime radix (such as 2,3,4,5,7,8,9,11,13,16,17,...)". However, 9 and 16 are not prime (but it is still true that they only have trivial automorphic numbers) 184.12.106.188 (talk) 08:19, 27 January 2016 (UTC)

Well spotted. It was a misunderstanding when shortening "Radixes which contain only one prime factor" in [2]. I have changed it to say "A prime power radix".[3] PrimeHunter (talk) 11:57, 27 January 2016 (UTC)

Spherical numbers

"The terms circular and spherical were formerly used for the slightly different case of a number whose powers all have the same last digit as the number itself, the only spherical numbers being 5, 6 and 10."

Obviously 1 should be added, and I will do this, even though it's a trivial case. But reading the definition strictly, it sounds like any number ending in 1, 5, 6 or 0 is spherical? E.g. all powers of 15 end in 5, which is the last digit of 15. 2.24.119.38 (talk) 10:01, 9 December 2018 (UTC)

"Higher order" automorphic numbers

Apart from automorphic and trimorphic numbers, there are also numbers where a higher than third power is the lowest which ends on the digits of the number itself. For example, the powers of 2 are 2, 4, 8, 16, 32, which means one could call 2 a "pentamorphic" number (2⁵ = 32). In case of 11, the 11th power is the lowest one to end with the digits "11". There are also numbers where this is never the case regardless of the exponent: For example, the perfect powers of 15 all end on either "25" or "75", but not "15".

Considering trimorphic numbers, there are also trivial cases in every positive integer (B) base system: 0 (0³ = 0), 1 (1³ = 1) and B-1 ((B-1)³ = B³ - 3B² + 3B - 1) always work. That would be 0, 1 and 9 (9³ = 729) in decimal. Strings of the highest single digit (99, 999, ... in decimal) seem to work as well, as seen in the examples for base-10 and 12. Plus, one could call automorphic numbers like 6 "trivial" trimorphic (and also tetramorphic, pentamorphic, ...) numbers, as their higher powers necessarily all end in the same digit(s) again.

Does this have a connection to some other field of mathematics? E.g. like the divisibility rules for prime numbers are related to Fermat's little theoreme?

After all, there are certain patterns. Like, automorphic numbers are also trimorphic, tetramorphic etc. Trimorphic numbers are also pentamorphic, heptamorphic etc. but not tetra- or hexamorphic (unless they are automorphic).

The most generalized form of "automorphic" numbers would be all natural numbers n for which the polynom a*n^b with a,b € N ends on the digits of n itself in a given integer radix. --2003:E7:7730:FF82:4189:4EE6:88C8:8C6F (talk) 17:07, 22 August 2020 (UTC)

(BTW, I've kind-of answered the previous question: All powers of 15 end in 5, but for 15 being a "polymorphic" number, some integer power >1 would need to end in the two digits "15". In decimal, this is not the case afaik.)