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WikiProject Mathematics (Rated Start-class, Low-importance)
WikiProject Mathematics
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
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 Field: Basics


What does:

Base 6 can be counted on hands by using each hand as a digit (0 to 5).

mean? I assume the second 'hand' is a mistake. Tompagenet 10:33, 5 Aug 2003 (UTC)

It was probably a presumption that you count 1-6 instead of 0-5... Dysprosia 10:34, 5 Aug 2003 (UTC)
What is probably meant is: Take your right hand, close it. This is "0". Open one finger, this is "1", etc. Open all five fingers, this is "5". So on one hand you can handle one "senary digit". With two hands you can count from 0 to 35 (= 55(6)). --SirJective 12:04, 21 Nov 2003 (UTC)
This is precisely what I meant Karl Palmen 15:23, 21 Nov 2003 UT

prime numbers[edit]

So this article implies, if p is prime, then p mod6 = 1 or p mod 6 = 5 ... can we get a link to an article or reference for this? linas 01:10, 1 Apr 2005 (UTC)

Dohh, never mind, its obvious. linas 00:38, 10 August 2005 (UTC)


All prime numbers besides 2 and 3 end in either 1 or 5. Did the definition of Prime Number change? What happened to only being divisible by 1 or itself? What about 13, 17, 19, 23, 29, 37, 43, 47, 53, 59, 67, 73, 79, 83, 89, 97, 103, 107, 109, 113, ...? malachid69 08:14, 9 Aug 2005 (UTC)

That's base 10 not base 6. The statement is that p mod 6 is congruent to 1 or 5. The article wording should be changed to say "congruent" exactly in order to avoid this confusion. linas 00:38, 10 August 2005 (UTC)

big errors[edit]

There is a big mistake in this page. No number ending in 5 (except for 5 itself) is a prime number. Every single number with 5 as the final digit can be devided by 5. A correct list of prime numbers would be 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, etc.

I also noticed that there are quite some known perfect numbers that don't have 44 as the final two digits, like 28, 496 and 8128.

I don't know who wrote this page, but his maths are worse than mine, and people should not try to write articles about things they don't know about.

I am not great at maths, so I have no idea how I should correct this error and I don't have a clue what Senary would be good for, but I just thought I'd write this down here so someone who does know can correct the error here and then delete my message.

(The above message was posted on the main article page by (talk · contribs). I'm just moving it here. - ulayiti (talk) 12:00, 20 August 2005 (UTC))

The poster didn't have a clue. However, everyone who reads this article seems to trip over this, so I will try to fix the article in just a moment. linas 15:56, 20 August 2005 (UTC)
OK, I think I fixed it for good. linas 16:15, 20 August 2005 (UTC)

consistency errors[edit]

I was looking through the other numeral systems, and I saw the box on the right that had all of the numeral systems in a single easy-to-use box, I would put it in myself, but i don't know how... example: Unary numeral system 18:37, 26 August 2005


This entry is almost incomprehensible for a layman. As a layman who knows nothing about numbers and number systems, maybe someone could edit this so that someone who doesn't know what base 6 is could understand it? Isn't that the point of Wikipedia? This entry is useless to someone with limited knowledge of math terminology, symbology. —The preceding unsigned comment was added by (talk) 00:58, 27 April 2007 (UTC).

A layman could click on numeral system or base early in the article to get information necessary for understanding the article. Karl 10:03, 27 April 2007 (UTC)

"Furthermore, all known perfect numbers besides 6 itself have 44 as the final two digits"[edit]

This means nothing. Every number (2n-1)2n-1 with n = odd number > 2 has 44 as the final digits in base 6. The even perfect numbers are only a part of these numbers. --Zumthie (talk) 23:46, 10 December 2008 (UTC)

So does every power of 10 besides ten itself. I disagree that such an argument makes the statement meaningless. Which other bases have all known perfect numbers besides 6 ending with the same two digits? Karl (talk) 11:33, 12 December 2008 (UTC)

See base 2:
even perfect numbers
base 2 110 11100 111110000 1111111000000 1111111111111000000000000 111111111111111110000000000000000 1111111111111111111000000000000000000
(22-1)21 (23-1)22 (25-1)24 (27-1)26 (213-1)212 (217-1)216 (219-1)218
base 10 6 28 496 8128 33550336 8589869056 137438691328
In base 2 the first part of an even perfect number is a Mersenne prime, digits 1, the rest of the digits are 0.

--Zumthie (talk) 23:00, 12 December 2008 (UTC)

55 base 6[edit]

556 ends in a 5 but is 3510 which is certainly not a prime number. (talk) 13:31, 15 March 2010 (UTC)

All primes end in 1 or 5, not all numbers ending in 1 or 5 are prime. — sligocki (talk) 04:03, 17 March 2010 (UTC)


Just wondering who proved that all prime numbers end in 1 or 5 when written in base 6 (or whether it is proved). The article states it without any sources or proof. A citation would also be useful for the claim that all known perfect numbers (when written in senary) end in 44. (talk) 11:18, 31 March 2010 (UTC)

It is analogous to the reason that no prime numbers in decimal end in 2, 4, 5, 6, 8. Here's a proof: any number in base 6 is of the form x = a_0 + a_1 * 6 + a_2 * 6^2 + a_3 * 6^3 + ... = a_0 + 6 * (a_1 + a_2 * 6 + a_3 * 6^2 + ...) = a_0 + 6k, where k is an integer (here a_n are the senary digits). Thus if a_0 is divisible by 2 or 3, then x = a_0 + 6k is divisible by 2 or 3. 1 and 5 are the only senary digits that are not divisible by 2 or 3, so all primes (other than 2 and 3) have last senary digit 1 or 5. Cheers, — sligocki (talk) 01:22, 1 April 2010 (UTC)
It is well known, that all even perfact numbers are the product of a Mersenne prime and half of one plus that Mersenne prime. The powers of four besides 1 end in either 04 or 24 in senary. This can be proven by induction. Doubling these numbers and subtacting 1, leads to a set of numbers that contains all Mersenne primes besides 3 and they end in 11 or 51 respectively. When the respective numbers are multiplied one always get a number ending in 44. 44 is an automorphic number in senary just as 76 is in decimal. Karl (talk) 12:14, 1 April 2010 (UTC)
Thanks. The proof should be included in the article though. (talk) 12:35, 13 April 2010 (UTC)

Senary calculator please[edit]

At "Fractions" section I see senary digits. Please, where I can find a senary calculator? Not a base convertor, a calculator. For example what is senary for 1.618...(phi in base 10)? :) Bigshotnews 19:01, 16 October 2010 (UTC) —Preceding unsigned comment added by Bigshotnews (talkcontribs)

The golden ratio begins 1.34125455435343145134223514... in base 6, or 1.m8yzrxravmena... in base 36 (compressed senary). Wolfram Alpha should work as a calculator in whatever base you want (just type "in base 6" or something similar at the end of your query). Double sharp (talk) 15:06, 29 January 2015 (UTC)

senary addition and multiplication tables[edit]

Hi editors,

I tried to add a link to, containing senary addition and multiplication tables, and a simple converter also. The link was deleted by Quiddity, see [[1]]

I still think it should be posted.

Yura.vrilyuk (talk) 16:15, 3 July 2013 (UTC)

I don't see a problem posting this link, though we do not want to end up with links to every converter calculator, etc. Bcharles (talk) 15:24, 22 July 2013 (UTC)

Merge from base 36[edit]

As base 36 is based on a six squared, it is essentially a notation for two senary digits. It would best be covered as a section of the senary article. Bcharles (talk) 03:52, 12 July 2013 (UTC)

This doesn't seem like a good idea as the two systems are sufficiently different that they deserve separate treatment.
  1. Base 36 notation represents the two senary digits by a single alphanumeric digit; in fact it's a notation for representing arbitrary numbers (which are usually not senary at all).
  2. Base 36 has a number of practical computer applications, while base six arose in totally different historical traditions.
In sum, combination doesn't seem to have any real advantages. --SteveMcCluskey (talk) 15:27, 19 July 2013 (UTC)
Both of these reasons point to "base36" encoding scheme rather than a base 36 numerical system. Perhaps it would be best to have "base 36" as an article focused on encoding with an "about (positional numeral system)" template linked to senary; similar to "base 64" with link to binary. Bcharles (talk) 20:52, 19 July 2013 (UTC)
If you operated that policy you may as well merge binary with hexadecimal!!! YOU CANNOT MERGE DIFFERENT BASES. Properties of base 6 like the ending digit of prime numbers DO NOT APPLY TO BASE 36. This is an awful idea. Bellezzasolo (talk) 20:50, 25 February 2014 (UTC)
They apply similarly. In base 6 all even perfect numbers end with "44", apart from 6 itself; thus they must end in "S" in base-36. In base 6 all prime numbers except 2 and 3 end in "[x]1" or "[x]5", so in base 36 they must end in 1, 5, 7, B, D, H, J, N, P, T, V, or Z. One place in base-36 exactly corresponds two places in base-6, and so such numbers can be converted easily. In fact, hexadecimal is today mostly popular simply because it has this relationship with binary! The only reason octal and hexadecimal are covered separately is that they're both popular enough in their own right. With base-36, which is large and thus obviously unpopular, a merge with its little sister base 6 seems more appropriate. Double sharp (talk) 15:24, 28 March 2015 (UTC)
Merged sections of base 36 into senary (the mathematical properties) and base36 (the compression scheme). Double sharp (talk) 17:26, 28 March 2015 (UTC)


I think the lead needs a major rewrite. In addition to what I just changed (using standard modular arithmetic notation), there needs to be a break separating the basic mathematical properties from the summary. I'm not sure the proof that all primes "end" in 1 or 5 should be in the article, but I'm sure it shouldn't be in the lead. — Arthur Rubin (talk) 19:01, 23 July 2013 (UTC)

I moved most of the lead into a section on 'mathematical properties', reordered the sections and rewrote the lead section. I am lost reading the standard modular notation. Can you add some brief explication. Bcharles (talk) 23:45, 24 July 2013 (UTC)

Base 36 Redirect[edit]

I'm not sure, but I think it would be more logical to redirect "Base 36" to Base36 rather than Senary#Base 36 as senary compression. WikiWisePowder (talk) 20:17, 9 November 2015 (UTC)