## Bug #1243

### 1 is prime

**Description**

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Prime.prime? always returns true for n < 2

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#### Updated by yugui (Yuki Sonoda) over 11 years ago

**Status**changed from*Open*to*Closed***% Done**changed from*0*to*100*

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Applied in changeset r22741.

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#### Updated by headius (Charles Nutter) over 11 years ago

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The number 1 is not considered prime:

http://en.wikipedia.org/wiki/Prime_number#Primality_of_one

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#### Updated by daz (Dave B) over 11 years ago

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- value = -value if value < 0 <--- NOT required ?
- return false if value < 2

I think your negative guard is not required?

All negatives, 0 and 1 are non-prime, therefore all < 2 are non-prime.

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

daz

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#### Updated by yugui (Yuki Sonoda) over 11 years ago

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On 3/4/09 1:40 AM, Dave B wrote:

- value = -value if value < 0 <--- NOT required ?
- return false if value < 2
I think your negative guard is not required?

For a arbitrary positive prime number p, -p is a prime element in the

ring of integers. On the other hand, it is sure that just saying "prime

numbers" means positive ones.

(http://en.wikipedia.org/wiki/Prime_number)

Without further specification, however, “prime number” always means a

positive integer prime

I am at a loss whether -2.prime? should return true or false? Which is

more useful for Rubyists?

--

Yugui yugui@yugui.jp

http://yugui.jp

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#### Updated by yugui (Yuki Sonoda) over 11 years ago

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On 3/4/09 1:29 AM, Charles Nutter wrote:

The number 1 is not considered prime:

Sorry, I wanted to mean that the implementation of Prime regarded 1 as a

prime.

--

Yugui yugui@yugui.jp

http://yugui.jp

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#### Updated by daz (Dave B) over 11 years ago

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Ring theory defines "prime element" as a special case.

Someone else might request adding a "prime_element?" or "ring_prime?" method, later.

Maybe they could use ( -7.abs.prime? ) for their specific application.

The ordinary definition of prime must return false for ( -2.prime? )

A (true) result for ( -2.prime? ) or any other negative numbers is wrong,

regardless of how useful it would be to Rubyists. ;))

daz

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#### Updated by RobertDober (Robert Dober) over 11 years ago

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On Wed, Mar 4, 2009 at 9:15 AM, Yugui (Yuki Sonoda) yugui@yugui.jp wrote:

On 3/4/09 1:40 AM, Dave B wrote:

- value = -value if value < 0 <--- NOT required ?
- return false if value < 2
I think your negative guard is not required?

For a arbitrary positive prime number p, -p is a prime element in the

ring of integers. On the other hand, it is sure that just saying "prime

numbers" means positive ones.(http://en.wikipedia.org/wiki/Prime_number)

Without further specification, however, “prime number” always means a

positive integer primeI am at a loss whether -2.prime? should return true or false? Which is

more useful for Rubyists?IIRC 1 was discarded as prime because it messed up the uniqueness of

factorization. If we want to keep this spirit I see no solution for

negative numbers.

One could maybe define that all negative numbers are composed with the

exception of -1, which is prime ARRRGH and the additional rule that -1

can only occur ? times (in the regexp sense of ? ) in a factorization.

But maybe it makes more sense to say x.prime? if and only if x.abs.prime?

and

for all x: x.primefactors.count{ | f| f.negative? } < 2

I guess these are lousy ideas, does anyone have something better to suggest?

Cheers

Robert

--

Yugui yugui@yugui.jp

http://yugui.jp

--

There are some people who begin the Zoo at the beginning, called

WAYIN, and walk as quickly as they can past every cage until they get

to the one called WAYOUT, but the nicest people go straight to the

animal they love the most, and stay there. ~ A.A. Milne (from

Winnie-the-Pooh)

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#### Updated by RobertDober (Robert Dober) over 11 years ago

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On Fri, Mar 6, 2009 at 8:42 PM, Jacob Fugal lukfugl@gmail.com wrote:

I have a hard time imagining a situation where asking if a negative

number is prime would be necessary. Code the cares about prime numbers

almost by design implies that it expects positive integers. In that

mindset, I call YAGNI.There is no compelling argument (to me) to have the Ruby standard

library definition of prime be inconsistent with the mathematical

definition. A prime number -- not a prime factor or a prime element of

the ring of integers or a prime element of any other arbitrary ring --

is a natural number (that is, a positive -- or non-negative, it

doesn't matter here -- integer) which has exactly two distinct natural

number divisors: 1 and itself. 1 is not prime and negative integers

are not prime. Rationals or Floats not coinciding with an integer are

not prime. That should be the definition of "prime?".In the rare case where someone does need to check if the absolute

value of a negative integer x is prime, x.abs.prime? is not a horrible

burden, and it reveals the intent that negative numbers are acceptable

input to boot.Jacob Fugal

I tend to agree, I was taken away by the "challenge" to find a

meaningful definition for primes < 0. And when I failed to do so I

asked for help, stupid somehow, is it not, LOL.

Cheers

Robert

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