WikiProject Mathematics (Rated Start-class, Low-importance)
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Mathematics rating:
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Field:  Number theory

## 189

So... unless I'm not getting the concept right, 189 is not a Harshad number, is it? Since 1 + 8 + 9 = 18, and 189 / 18 = 10.5 , which is not an integer. What am I missing? — Preceding unsigned comment added by 213.61.58.164 (talk) 09:53, 13 February 2012 (UTC)

You are correct. I don't know why 189 was listed but I have removed it. —David Eppstein (talk) 15:59, 13 February 2012 (UTC)

## Error in statement of theorem

The article says:

H.G. Grundman proved in 1994 that in base 10 there are no sequences with more than 20 consecutive Harshad numbers,

but there is a missing word here, since clearly there is a sequence with more than 20 consecutive Harshad numbers, namely the sequence of Harshad numbers, whose initial segment is cited at the top of the article. Perhaps this should say "arithmetic sequence"? -- Dominus 00:53, 9 May 2004 (UTC)

### No error

I just misunderstood it. The article was apparently saying that there are no sequences of more than 20 consecutive numbers that are all Harshad numbers. I have reworded the article in a way I find clearer. -- Dominus 00:58, 9 May 2004 (UTC)

I find your way to be clearer, too. Thanks for clarifying it. PrimeFan 19:57, 9 May 2004 (UTC)
Ditto. I was trying to fix it, but realized I wasn't sure what it originally meant. Thanks. Grendelkhan 22:35, 2004 May 9 (UTC)

It is my pleasure to assist. -- Dominus 00:34, 10 May 2004 (UTC)

## Understatement of the century

He [Grundmann] also found the smallest sequence of 20 consecutive integers that are all Harshad numbers; they exceed 1044.

The above statement is correctish, but at the same time looks like a misinterpretation of the facts. Various sources claim that the 20 numbers in the sequence exceed 1044363342786. I wasn't able to find more information, which is why I'm not editing the article right away.
Herbee 22:43, 2004 May 12 (UTC)

MathWorld agrees, so i changed the article Illuvatar 17:16, 7 Sep 2004 (UTC)

## Inconsistency

Is it Harshad or Harshard? Both appear multiple times in the article.

• I changed them to Harshad. It's the name of the article, it gets more google hits, and it shows up in mathworld. If it can also be spelled Harshard, we should mention it

## Origin ?

Why is it called a Harshad number ? Is Harshad name of a person ? Jay 07:06, 10 Nov 2004 (UTC)

## Factorials in base 10

Anyone know why the factorials are all Harshad numbers? Anyone know why it's only in base 10? --Doradus 19:52, Nov 11, 2004 (UTC)

Not all are: the article says 432! is the first which is not. But as an aside, factorials can be dived by a large number of numbers, while the higher the base, the smaller the sum of the digits, and so the more likely they are to divide into a factorial. --Audiovideo 17:35, 23 March 2007 (UTC)

## Consecutive Harshad Numbers -- Generalization

Base 2 -> Infinitely many sequences of 4 consecutive numbers. Base 3 -> Infinitely many sequences of 6 consecutive numbers. ... Base 10 -> Infinitely many sequences of 20 consecutive numbers? [Article doesn't say it, but implies it, IMHO.] Does anyone know if this can be generalized? 70.178.215.64 11:08, 7 January 2006 (UTC)

It certainly says it now. PJTraill (talk) 00:55, 24 January 2014 (UTC)

## Notability

I can see this has enough notability for an article, but does it have enough to be linked from each Harshad number. I don't see it as a notable fact about 300 or similar numbers --Audiovideo 17:38, 23 March 2007 (UTC)

Not really, since it only applies to base 10 Harshad numbers. PJTraill (talk) 23:34, 23 January 2014 (UTC)

## More Numerology than Number Theory

Although the article correctly identifies this property of "numbers" as base specific, I think it should explicitly point out the this is a property of number representation and not of numbers themselves. Any number is a Harshad Number if you choose the right base. Pure properties of numbers, e.g. primeness, are base independent. —Preceding unsigned comment added by 86.153.60.15 (talk) 17:48, 26 January 2009 (UTC)

Not numerlogy, which is superstition, also not true number theory, just recreational mathematics, which is harmless. It is a property of number pairs. See further my reaction a few minutes ago in a later section. PJTraill (talk) 23:55, 23 January 2014 (UTC)

## Is this statement correct?

This sentence at the end of the introduction seems to contradict the list of Harshad numbers" "All integers between zero and n are Harshad numbers." Has a qualifying phrase been removed about bases or something? - DavidWBrooks (talk) 13:11, 5 March 2009 (UTC)

Aha! the phrase "... in base n" had been truncated, but is now back. I figured it was something like that. - DavidWBrooks (talk) 15:37, 5 March 2009 (UTC)

## A statement which appears to be problematic

The article says: "Interpolating zeroes into N will not change the sequence of digital sums, so it is possible to convert any solution into a larger one by interpolating a suitable number of zeroes" According to this statement, since 112 is a Harshad number in base 10 (1+1+2=4, and 122 is divisible by 4), so would 1102 - but it clearly isn't, since no umber ending in 02 is divisible by 4. 77.125.4.148 (talk) 06:36, 29 March 2011 (UTC)

The catch is in the word “suitable”; I shall try to clarify that. PJTraill (talk) 00:56, 24 January 2014 (UTC)
I have done that, but I was not able to provide a complete proof; perhaps someone can add the remaining details.PJTraill (talk) 03:18, 24 January 2014 (UTC)

## Use and Purpose

One thing I have always found with many of mathematics articles is a lack of information on relevance and uses. Coming from a science and engineering background, I see math as a tool, with a means to obtain expected results. I understand there are lots of mathematical terms that have no real basis in the world (or none currently discovered). I think it would be very useful and meaningful to include a section on application of the theories in the wiki, even if the application is something as basic as: "Recreational Mathematical Artifact or exercise with no known engineering application."

I do not know enough about Harshad numbers, but could see potential in cryptography that may go beyond recreational purpose. fter reading through the article, though, the only function I gleamed was that of recreational mathematics. If someone knows an application, please take a moment to write something up. It would be great to know if the theory is purely for entertainment purposes, a process searching for some special meaning, or something with proven real world applications. — Preceding unsigned comment added by 216.55.51.54 (talk) 22:21, 3 April 2012 (UTC)

I agree, and this may also be related to the talk section "More Numerology than Number Theory". While Harshad numbers (particularly base 10) may be interesting playthings that bring joy to some folks, presumably they earned a name and continued interest because they are useful in some way. The only thing I've learned about Harshad numbers is that apparently a great many phone numbers are Harshad numbers. What that has to do with anything, I don't know. 24.57.218.21 (talk) 19:34, 26 June 2012 (UTC)
I googled Harshad numbers, and found nothing about applications, not even within pure mathematics. This suggests it is purely recreational mathematics (not numerology, if you follow Wikipedia in taking that to mean superstition or pseudomathematics). The purpose of recreational mathematics is to challenge and entertain, without the requirement in normal mathematics of significance; that is hard to define, but would include universality (does not depend on how many fingers we have or how we choose to write numbers) and applicability to other areas of mathematics or the the real world; as a result of applicability, normal mathematics also builds up considerable depth: to prove a result, many others may be required. Lack of significance reduces the satisfaction, but makes the problems approachable without training. Harshad numbers thus appear to be useful only insomuch as they entertain; it would be unfair to scoff at (added: such PJTraill (talk) 18:07, 24 January 2014 (UTC))articles, but reasonable to expect it to be made clear. (I see it asserted elsewhere that the phone number business is nonsense, which sounds likely.) PJTraill (talk) 23:51, 23 January 2014 (UTC)

## Niven numbers were referenced as early as 1984 and base characteristics proven first by Robert kennedy and Curtis cooper

This is a good overview of the actual history with proper credit...

1985 paper...

http://www.hindawi.com/journals/ijmms/1985/386824/abs/