When was 1 a prime number?



#53

In my first response to Bill's Bowling Challenge Variation challenge I used 1 as a prime number. Don Shepherd noted that 1 isn't a prime number, but Katie came to my rescue by noting that 1 had been considered to be a prime number in the past.

Even so, I wasn't sure what I might have relied on from the olden days to accept 1 as a prime number. Then I found that Table 24.9 on page 870 of my copy of AMS-55 published in 1964 didn't include 1 in the list of prime numbers. But yesterday I was looking up something else in my copy of Mathematical Tables from the Handbook of Chemistry and Physics (1959). I looked at the table of factors and primes starting on page 242. A note at the top of the table states "If n is prime the mantissa of its logarithm is given." The mantissa is given as 0000000 for 1.


#54

Quote:
When was 1 a prime number?

In hindsight it never was. However, according to http://en.wikipedia.org/wiki/Prime_number#Primality_of_one, the 18th century, 1956, and 1941 (death of the last believer).

#55

Always was, never was, depending on how you defined "prime number". These days it's mostly not, but the question shows up sometimes when non-professional people are asked about prime numbers. We had this just last week at our health club, the daily question was "What is the sum of the first five prime numbers?" and the "correct" answer was ( 1 2 3 5 7 11 ) either 18 or 28, depending on whether you started with 1 or 2.

(The definition "a natural number divisible, without remainder, only by itself and one" leads to the conflict; there's an implication that that the word "and" means "itself <> one" but not everyone sees or agrees with that.)


Edited: 30 Aug 2010, 1:18 a.m.


#56

The definition of prime number that the school textbooks use today is "a number with exactly 2 factors, 1 and itself", and a composite number is "a number with more than 2 factors". Since 1 has only 1 factor, it can't be prime, nor can it be composite.


#57

Quote:
The definition of prime number that the school textbooks use today is "a number with exactly 2 factors, 1 and itself",

Don,

Shouldn't this be "a number with exactly 2 distinct factors, 1 and itself"? Otherwise one could argue that 1 is prime because 1 = 1 * 1. I was taught composite numbers are those that can be expressed as a finite product of prime numbers. This would exclude 1 from the set of prime numbers because 6, for instance, could be expressed by the infinite product 2 * 3 * 1 * 1 * ...

Gerson.


#58

Gerson, I see what you are saying, but I don't know that you need the word "distinct." I think the definition works because 1 only has 1 factor, which is 1 (you can't count 1 twice), and a prime number needs to have exactly 2 factors, and 1 doesn't.

We teach it that way. We teach that both 0 and 1 are neither prime nor composite, they are special cases. I'm sure kids couldn't care less.


#59

IMHO, "exactly 2" implies a quantity with no mention of uniqueness. Distinct implies uniqueness. I know I can find a person on the street that will insist that 1 factors into exactly 2 numbers. 1 x 1. Because, most children spend a lot of time on a x b = c and remember that as adults.

From Wikipedia:

A prime number (or a prime) is a natural number that has exactly two distinct natural number divisors: 1 and itself.


#60

Hey Egan.

Well, I have no doubt that you could find a person on the street who would agree to almost anything; that doesn't mean they are right. I'm sure there are some today who would say man never landed on the moon. And, please everyone, let's not steer this thread into an examination of did man, in fact, ever land on the moon!

The Wikipedia definition is fine. Middle school textbooks typically don't say "distinct", but that's OK, most kids are able to correctly distinguish prime and composite numbers, at least long enough to demonstrate that knowledge on a test. And, believe me, kids have no interest in debating whether or not 1 is prime.

I'd bet that if you took a survey of people on the street and asked what is the difference between a prime and a composite number, maybe 5% would know. Things like that are just not relavent in people's lives.

Edited: 30 Aug 2010, 3:19 p.m.

#61

Quote:
Gerson, I see what you are saying, but I don't know that you need the word "distinct." I think the definition works because 1 only has 1 factor, which is 1 (you can't count 1 twice), and a prime number needs to have exactly 2 factors, and 1 doesn't.

Don,
So, does that mean that 12 has only 2 prime factors? ... I think you'll confuse the kids greatly if you don't say when duplicates matter.

Regarding the primality of 1, I think the key thing is that defining it as non-prime makes some other mathematical concepts work out better. I wish I could remember what those concepts were though.... :) This is an example of a key concept that I think should be taught explicitly in school - that sometimes in math we define things a particular way to make the model work better or easier.


#62

Well, 12 has 6 factors, which makes it a composite number (>2 factors). That's as far as middle school kids get into this subject.

The non-primality of 1 (and 0, for that matter) is just something we don't get into with these kids. And most of them understand which numbers are prime and which are not. And they LOVE making factor trees, which is nice.

Then we move from prime/composite into greatest common factor and least common multiple, which of course are used for simplifying and adding/subtracting fractions with different denominators.

But by the time they graduate high school, they've probably forgotten it all. Which, honestly, doesn't disturb me. They've exercised their minds and learned how to think logically, and that's really the important part.


#63

Don:

You wrote:

Quote:
But by the time they graduate high school, they've probably forgotten it all.

That pretty much agrees with my experience. Somewhere in my early education I was exposed to prime numbers, Mersenne numbers, perfect numbers, et al, but in forty years of work with military electronics systems I can't recall a single instance in which I had any use for any of them. Is work with these numbers primarily limited to mathematicians? Are there engineering disciplines depend on them in some way?

Palmer


#64

Quote:
Is work with these numbers primarily limited to mathematicians? Are there engineering disciplines depend on them in some way?

Palmer, I can't speak for the engineering disciplines regarding their possible use of prime numbers, but I do know that the entire security system behind the Internet is based upon the impossibility of factoring certain very large numbers into their two prime factors. I believe this is the RSA public key encryption system. I'm certainly no expert in that, but I do tell my students that if they can figure out a way to factor huge integers, they just might put the Internet out of business or, more likely, earn a few million dollars by agreeing to keep their discovery secret!

RSA at one time offered lots of money to anyone who could factor certain large numbers; I believe they discontinued that. I printed out one of those numbers and showed it to my students and told them how much they could win if they could factor it. One student actually spent the weekend trying to divide the number by 2, and when he reached the last digit he realized that there would be a remainder. On Monday morning he brought in his work, and I remember the really bad feeling he had when I pointed out that he could have saved himself all that work if he had only realized that the number ended in 3 and therefore could not be divisible by 2!

#65

Don, do you teach Euclid's algorithm to find the greatest common factor? I think it would be a great way to show kids the joy of a good algorithm and a nice way to introduce them to the concept of programming. I'm not suggesting that they would write a program, but you could explain the algorithm and then say "this is the sort of step-by-step procedure that can be done by a computer."


#66

For the sixth graders, we generally teach it this way. For the seventh and eighth graders (if it is included in their curriculum), I might teach them Euclid's algorithm like this if I think they are capable of understanding it. With some classes, the simplest method is the best, but a good teacher challenges inquisitive students with methods they are capable of understanding and using. I try to be a "good teacher," of course.

It is much easier to tailor your instruction to the individual student if you have 13 students a day, as opposed to 150 students a day. Fortunately, I have 13.

#67

I disagree with the belief that 1 isn't a prime number the same as I disagree with Pluto not being a planet.

However, "exactly 2 factors" means 2 and no more or less. 1 = 1 * 1 is two factors but not "exactly" 2 factors because 1 = 1 * 1 * 1 also.

However, I think that definition is also wrong. Definition changes are done for one reason and that reason is to try and exclude 1, for no particular purpose. 1 being a prime violates nothing in mathematics and fits the old definition of a prime number.


#68

Mike, you are free to believe what you like, of course. But the reality is that 1 is not a prime number, and that's what kids are taught.

#69

Ultimately, 1 is defined as non-prime to keep a pattern intact, according to Dr. Math:

http://mathforum.org/library/drmath/view/57058.html


#70

I've been staying out of this since my initial comment on Palmer's use of 1 as a prime number.

But if you want to debate something more relevant to HP calculators, why does the 20b and 30b define 0^0 as 1? I think that this is wrong but there are plenty of examples of this, try the Google calculator.


Edited: 1 Sept 2010, 12:55 p.m.


#71

For 0^0 the 33s returns INVALID yx, the 48GX returns 1.The HP 50g returns ?, that is, a mathematical indetermination, however for 0.^0. it returns 1. Perhaps there's a practical reason for this behavior, but I can only guess.

#72

Maybe this sheds some light (from Wikipedia):

Quote:
Treatment on computers

The IEEE 754-2008 floating point standard is used in the design of most floating point libraries. It recommends a number of different functions for computing a power:[19]

* pow treats 0^0 as 1. This is the oldest defined version, it checks if the power is an exact integer and uses the value defined by pown in that case otherwise the value is as for powr except for some extra exceptional cases.
* pown treats 0^0 as 1. The power must be an exact integer. The value is defined for negative bases, e.g. pown(-3,2) is 9.
* powr treats 0^0 as NaN (Not-a-Number - undefined). The value is also NaN for cases like powr(-3,2) where the base is less than zero.

Programming languages

Most programming language with a power function are implemented using the IEEE pow function and therefore evaluate 0^0 as 1. The later C[20] and C++ standards, and the Java standard[21] mandate this behavior. The .NET Framework method System.Math.Pow also treats 0^0 as 1.[22]



#73

There's a good, short discussion of the issue here. The vast majority of HP calculators (and I think all of them from the Pioneer series backward) return an error for 0^0 as do most other calculators. I wonder if there's much thought given to this or whether it's just a matter of convenience as to what the calculator returns.

Just curious, has TI been consistent over the years as to how they handle this? (I don't have enough TI calculators to check.)


#74

Quote:
Just curious, has TI been consistent over the years as to how they handle this?

 TI-82: ERR:DOMAIN
TI-83+: ERR:DOMAIN
TI-85: ERROR 04 DOMAIN
TI-86: ERROR 04 DOMAIN
TI-89: 1.
TI-92+: 1.

#75

Some more TI's:

NSpire CAS     undef
34ii error
34 Multiview error
30XIIS error
30XS error
36X error
15 error

Sharp EL531W gives error (the only calculator I've seen with a built-in base 5).

Edited: 1 Sept 2010, 8:27 p.m.


#76

It seems like TI and HP have been following a similar path, I think it pretty strange (and a bit disturbing) that both companies chose to abandon consistency on this with some of their recent high-end calculators. Pretty soon they'll start returning answers when you divide by zero like some of the earliest calculators did.


#77

Quote:
Pretty soon they'll start returning answers when you divide by zero like some of the earliest calculators did.

Actually on the HP 48/49/50 this is achieved by setting flag -22.

Edited: 1 Sept 2010, 8:46 p.m.

#78

Quote:
the only calculator I've seen with a built-in base 5

An interesting feature for former abacus users?

http://en.wikipedia.org/wiki/Quinary

#79

Katie:

You wrote:

Quote:
Just curious, has TI been consistent over the years as to how they handle this? (I don't have enough TI calculators to check.)

Here are some results from older TI's:
TI Business Analyst I  1
TI-30 1
SR-40 1
TI-55 1
TI-59 1
TI-68 Error
TI-80 ERR:DOMAIN
TI-81 ERROR 04 MATH
TI-83+ ERR: DOMAIN
TI-85 ERROR 04 DOMAIN
TI-86 ERROR 04 DOMAIN
TI-89 Titanium undef
voyage200 undef
where my result for the TI-89 Titanium is different from the result reported by Gerson. I don't know how to explain the difference unless there is a difference between the TI-89 and the TI-89 Titanium. I get the "undef" result whether I am in AUTO, EXACT, or APPROXIMATE. Using my results there is a consistency of sorts. Before the TI-68 the result is 1, and from the TI-68 forward the result is an error.

Here is an interesting one from the HP-19BII: ERROR: 0^0

where it has separate error messages for 0^NEG, 0/0, 0^0, and /0.

Palmer

Edited: 1 Sept 2010, 9:55 p.m.

#80

0^0 = 1 is somewhat logical if you look at the graph of y=x^x, and then take the limit as x->0 from the right. Obviously the limit is 1. Weird #$@%$#!! happens from the left, but I'm comfortable with the right-hand limit value 95% of the time. :)

#81

Quote:
"a number with exactly 2 factors, 1 and itself",

Factor 1: 1
Factor 2: itself

Factors imply multiplication

1 * itself = 1

IF one is not a prime, then the definition was changed along the way. The definition I was taught (in olden times) was "a prime is a number that is divisible only by itself and one". And I'm sticking with that, unless someone asks me to bet on that sum of first five primes question.

Edited: 31 Aug 2010, 7:19 p.m.


#82

One is not prime today. It and zero are neither prime nor composite. That's what we teach the kids.


#83

Yeah! But what we teach kids, isn't the the criteria for deciding whether or not 1 should be prime. It's not a prime today, for only one reason... to be consistent with a new definition.

Mathematics has been around for eons. Up until a hundred years ago, and even into the 20th century, many professional mathemiticians considered 1 a prime. Up until the 50s and even in the 60s when I was in school, '1' was a prime number.

What has changed is a definition; not mathematics.

Hell, "what we teach kids" is not only often wrong, but irrlevant. It wouldn't surprise me that some history revisionists "teach" that the U.S. lost the Vietnam war and offer as proof, the helicopters landing on the embassy in Saigon. Nevermind that the U.S. had alread been out of Vietnam for 2+ years and the war ended in a peace agreement.

Edited: 1 Sept 2010, 9:42 p.m.


#84

Quote:
But what we teach kids, isn't the the criteria for deciding whether or not 1 should be prime.

I agree. In 2010, 1 is not regarded as a prime number. That is the mathematically-accepted truth. We teach kids the mathematically-accepted truth.

Quote:
"what we teach kids" is not only often wrong, but irrlevant.

You've got it partially right. We don't teach kids "wrong" things. We teach the curriculum, and the curriculum is defined by the state educational organizations. Now they are not perfect, but at least where math is concerned I can't think of a single fact that we teach our kids that is "wrong."

Some of what we teach is, in my opinion, irrelevant. Box-and-whiskers plots and stem and leaf plots are examples. Kids will never see one of these once they leave middle school. But, truth be told, most kids will never see a quadratic equation once they leave high school either. But we teach them, because they make the kids think, and that's a value that never goes out of style.

#85

Quote:
Always was, never was, depending on how you defined "prime number".

I was taught that a prime was a positive integer that was only divisible by 1 and itself. 1 does fit that definition. And there may have been a period of time in my youth that I also believed it. The missing term distinct I read from a book later on. I believe that confusion is still being spread because that term is being dropped from casual (and perhaps formal) conversation.

1 is neither prime or composite--it is truly the loneliest number. 1 does have the unique distinction of being the unit, i.e. unit = 1/unit (or unit == 1/unit for TW, et al :-).

Edited: 30 Aug 2010, 10:40 a.m.

#86

Palmer, in my opinion defining 1 as non-prime is similar to defining 0! = 1. It seems to be an arbitrary way to get the math to work.

Regards,

John


#87

One reason why 1 is not a prime is that it would be in conflict with the Fundamental Theorem of Arithmetic.


#88

This is elegantly expressed in the video found here:

http://www.mathacademy.com/pr/#1

#89

I must be in a disagreeable mood today.

How does 1 violate the Fundamental Theory of Arithmetic? If what you say is true, then making 1 a prime number would violate that theorem. But if 1 was prime, it fits perfectly with the definition of Fundamental Theory of Arithmetic.

Fundamental Theory of Arithmetic

"In number theory and algebraic number theory, the Fundamental Theorem of Arithmetic (or Unique-Prime-Factorization Theorem) states that any integer greater than 1 can be written as a unique product of prime numbers."

In fact, in my view if that is the definition, you it is flawed without 1 being a prime.

For instance, according to that theory, what is the factors of 2?

2 must have uniquie product of prime numbers.

2 = 2 * 1

According to the theory, those factors must be prime. In order for that theory to hold, 1 must be a prime. If 1 is not a prime number, then there is no product of factors at all for 2. Product implies multiplication of 2 numbers.

The problem I see, is that all of these definitions are designed to exclude 1 from the list of primes, for no particularly good reason. The definitions are flawed. 1 being a prime voilates no rules of mathematics that I know of, unless it's related to a flawed definition.

So here is a test

1) What is the oldest known list showing 1 as a prime? (no definitions please; just lists of numbers)

2) What is the oldest known list showing 2 as the first prime? (no definitions please; just lists of numbers)

3) Which is older?

I know "todays" answer as to whether or not 1 is prime.
I know when I was going to school, many, many moons ago 1 was prime.
I also know that when I was going to school, Pluto was a planet.

From where I sit now; nothing has changed but definitions.

Edited: 1 Sept 2010, 9:43 a.m.


#90

Mike:

You asked

Quote:
1) What is the oldest known list showing 1 as a prime? (no definitions please; just lists of numbers)

2) What is the oldest known list showing 2 as the first prime? (no definitions please; just lists of numbers)

3) Which is older?


If you look at the first item in this thread you will see that my AMS 55 of 1964 does not show 1 as prime.

My Mathematical Tables from Handbook of Chemistry and Physics does show 1 as prime. It is fourth printing of the eleventh edition with the first printing in December 1959. The book was originally copyrighted in 1931.

In its definitions of "prime' my Webster's Collegiate Dictionary (Fifth Edition 1946) states

Quote:
5. math a. divisible by no number except itself or unity; as, 7 is a prime number. b. Having no common divisor but 1; -- used with to; as, 12 is prime to 25.

My 1969 issue of Encyclopedia Britannica says

Quote:
PRIME NUMBER, a positive integer (whole number) greater than 1 that cannot be expressed as the product of two positive integers neither of which is one.

Those are the oldest references that I have here in North Carolina. I have some older references in Florida including the "Mathematical Tables from ..." that I purchased in 1946 when I was a freshman in college. I also have the algebra and geometry texts that my father used in about 1918 where my memory says that the copyrights were in the very early 1900's. I will be able to look at those in about two weeks.

I told my wife that those old books would come in handy sometime!

Palmer

#91

Quote:
Palmer, in my opinion defining 1 as non-prime is similar to defining 0! = 1. It seems to be an arbitrary way to get the math to work.

There's been much parsing in this thread about the reasons for or against the primality of 1, but the fundamental point seems clear to me:

The purpose of integer factorization is to decompose an integer into a product of factors (smaller integers). A prime factor is one that itself cannot be decomposed. Any number -- integer or not -- can be expressed as the product of itself and unity (1), which renders 1 a "sub-factor" that cannot be used for decomposition. Thus, 1 would be trivial as a prime number.

It follows that two (2) cannot possibly be broken down, due to an absence of any smaller factors. However, to divide a positive even number by 2 will render an integer quotient that is smaller than the original number, so 2 is a valid factor that is prime.


As for the factorial of 0 (0! = 1): Yes, it seems arbitrary, but it fits the inductive mathematical definition and is useful in practice:

1! = 1        (by definition)
k! = k(k-1)! {k is integer}

--> (k-1)! = (k!)/k

Let k = 1,

--> (1-1)! = (1!)/1
--> 0! = 1/1 = 1
--> (-1)! = (0!)/0 = 1/0 (undefined)

-- KS


Edited: 2 Sept 2010, 1:50 a.m.

#92

I can verify that 1 is not prime from the highest authority.

TI-89 Titanium isprime(1) = false
TI-NSpire isprime(1) = false

Surely this should convince everyone.

: )
big smiley

Don


#93

HP 50g:

1 ISPRIME? -> 0

Now, I am convinced! :-)

Edited: 31 Aug 2010, 9:23 p.m.

#94

And I wager the algorithm is different for '1' than any other number. I bet the algorithm is something like:

isprime (n)
{
if (n==1)
{
return "false"
}
is_modernconvention_prime(n);
}

It's simply one extra step to the actual algorithm, to eliminate '1' by convention and nothing more.

I also bet you think Pluto isn't a planet?


Edited: 1 Sept 2010, 9:52 p.m.


#95

Mike, slavery is also no longer accepted in the United States.

Things change.

If you want to consider 1 prime or Pluto a planet, hey, it's a free country. But I've got to teach prime numbers to kids, and I want to teach the truth, and the truth is 1 is not a prime number. Maybe it once was, fine, but today it's not.

If you really feel strongly about this, perhaps you should write the American Mathematical Association and plead your case. I wish you luck.

Don


#96

I think I understand both viewpoints. There appear to be valid reasons why today 1 is defined as not prime, whether we* understand them or not; that is mathematical convention today, so that is what is taught in the schools.

However, "convention" is not the same as "truth". As David Hayden said:

Quote:
...sometimes in math we define things a particular way to make the model work better or easier.


*By "we", I mean "me". To me, it seems intuitive that 1 is prime. However, intuition often fails to maintain rigor in mathematics.


#97

It seems that the natural numbers are divided these days into four classes: 0; 1; primes; and composites. Glad I'm not a mathematician, and I still call Pluto a planet. Wrong, wrong.

Anyway, MathWorld has a discussion of this, with dates and sources. It's one of those arbitrary "math things". Primacy of One, and Primes

The Fundamental Theorem thing doesn't do it for me. If N is a prime > 1, then it has two factors, itself and one, and is the product of those two primes. But one is not a prime, so the theorem seems to fail. I suspect this is one of those "don't do math in English" problems, which I'm frequently reduced to in attempts to explain quantum and relativity physics.


Edited: 1 Sept 2010, 11:50 p.m.


#98

It's just occurred to me that the Eratosthenes Sieve would not work if 1 were defined as prime. In that case all multiples of 1 (that is, the whole set of integer numbers but 1) would be striken from the list and 1 would be the only prime number. Perhaps 1 should be called a primary number to distinguish it from the primes, but I'm not sure this is another Mathematics reserved word.

Including 1 in the set of prime numbers would spoil the mnemonic for the first seven primes in your reference. Likewise the exclusion of Pluto from the planetary family has ruined "My Very Educated Mother Just Served Us Nine Pizzas" :-)


#99

Quote:
Likewise the exclusion of Pluto from the planetary family has ruined "My Very Educated Mother Just Served Us Nine Pizzas" :-)



Ah, very interesting - in German it's "Mein Vater Erklärt Mir Jeden Sonntag Unsere Neun Planeten", so the loss of Pluto is even worse since both the number nine and the word "planet" are explicitely mentioned. ;-)



And what about other languages? How do you memorize the planets in French, Spanish or in other parts of the the world?



Dieter

In Portuguese it is, or used to be, "Meu velho terno marrom joguei sábado último no porão" which means "Last saturday I threw my old brown suit in the basement".

P.S.: I don't know if this is widespread as I cannot find any reference on the web. I rembember my father told it to me once (not every Sunday :-)

Another one, which takes into account the exclusion of Pluto, is "Meu Velho Tio Me Jurou Ser Um Netuniano", that is, "My old uncle has sworn me to be from Neptune". But who needs this to memorize only eight planets anyway?

Edited: 2 Sept 2010, 5:48 p.m.

Hi all,

The one I was taught in french:

"Mais Viendras-Tu Manger Jeudi Sur Une Nappe Propre ?"

and when Pluton was demoted, this was changed to:

"Mais Viendras-Tu Manger Jeudi Sur Une Nappe ?"


Which respectively mean:

"Will You Come To eat Thursday On a Clean Tablecloth?"

and

"Will You Come To eat Thursday On a Tablecloth?"

When Pluton was dropped, the tablecloth went dirty...

:-)

Etienne

Edited: 3 Sept 2010, 6:20 a.m.


Salut Etienne!

It appears these are not so difficult to create. Par éxample: "M'amie Voulait Toujours Me Jeter Sur Une Noce". However, does this one make any sense? :-)

Gerson.


Dear Gerson,

Well...it would make sense if my Grandmother could "throw me on a wedding" :-)

Even though she passed away a couple of years ago, she lived long enough to meet and appreciate my wife-to-be.

So...no question "she would have thrown me on a wedding" :-)

All the best

Etienne


The following isn't any better:

"Mers Vertes, Tremblez Maintenant ! Je Suis Un Navigant !"

Also, there is rhyme but no rhythm. I quit, this is not so easy as I imagined ...

The best so far is the German one, which is related to the (planets) subject.

Regards,

Gerson.


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