On 27 March 2018 at 09:26, Vincent Delecroix <20100.delecr...@gmail.com>
wrote:
>
>
> On 27/03/2018 10:05, Simon King wrote:
>
>> Hi Ralf,
>>
>> On 2018-03-27, Ralf Stephan <gtrw...@gmail.com> wrote:
>>
>>> sage: (2^23+1)/3
>>> 2796203
>>> sage: _.is_prime()
>>> False
>>> sage: factor(2796203)
>>> 2796203
>>>
>>> It turns out that Rational.is_prime does not exist and the fallback gives
>>> false answers.
>>>
>>
>> My first association: We should change the printed form of a rational
>> number. The integer 2796203 should print as 2796203, but the
>> corresponding rational number should be printed as 2796203/1.
>>
>> In that way, if a user copy-and-pastes printed output back into
>> Sage, there won't be a confusion between integers and rationals. >
>> I know that it is hardly possible to achieve that pasting of printed
>> output will ALWAYS create a copy of the printed object. However, in
>> cases where it is easily possible (the real number 1 should print as
>> 1.0, the rational number 1 should print as 1/1, the complex number 1
>> should print as 1.0+0.0*I etc), we should do it.
>>
>
> In plenty of CAS/python packages the difference is indeed explicit and I
> think that it is helpful to understand what is happening. And it would
> be helpful that copy/paste is working more consistently for basic Sage
> types.
>
> Note that a real number is not a good concept in computer science
>
> sage: CDF(1)
> 1.0
> sage: RealField(5)(1)
> 1.0
> sage: RealField(6)(1)
> 1.0
>
> and we also have other rings to treat like AA, QQbar, number fields,
> polynomials...
>
> sage: QuadraticField(2).gen()
> a
> sage: QuadraticField(-1).gen()
> a
>
> sage: ZZ['x']('x')
> x
> sage: ZZ['x','y']('x')
> x
> sage: QQ['x']('x')
> x
> sage: SR('x')
> x
>
> So +1 for this idea but it should be carefully designed.
>
>
> My second association: The fallback of is_prime should print a clear
>> warning
>> in the case that the input's parent is a field. Such as:
>> WARNING: The given number is defined as element of <self.parent()>,
>> which is a field. There are no prime numbers in a field, so, probably
>> you want to convert the number into <self.parent().ring_of_integer
>> s()>.
>>
>> It might be good to add an optional argument to the method that allows to
>> switch the warning off.
>>
>> Then, I tried to print a list of primes of the above form, using the
>>> global
>>> is_prime:
>>>
>>> sage: for n in range(1,100):
>>> ....: if is_prime((2^n - (-1)^n)/3):
>>> ....: print((2^n - (-1)^n)/3)
>>> ....:
>>> sage:
>>>
>>> No output. Turns out `is_prime(ZZ((2^n - (-1)^n)/3))` works. Really? How
>>> long does Sage exist without a fix to that?
>>>
>>
>> I suppose you are aware that from a maths point of view it isn't a bug,
>> since your code is (implicitly!) asking for prime numbers in QQ.
>>
>> However, I agree that it is (for most users) unintended behaviour, and
>> as a corollary to "explicit is better than implicit", we have "implicit
>> is not as good as explicit".
>>
>> I believe, when mathematically sound behaviour of SageMath is very likely
>> not what a user wants, then printing a fat warning that can optionally
>> be switched off (as above) is the correct fix. Just don't raise an error,
>> and please do not change the function to give a mathematically wrong
>> answer
>> ((3/1).is_prime() returning "True" would be wrong).
>>
>
> On the other hand we have some compatibility between ring of integer and
> fraction field such as
>
> sage: (3/40).factor()
> 2^-3 * 3 * 5^-1
> sage: (4/3).gcd(2/9)
> 2/9
> sage: (4/3).lcm(2/9)
> 4/3
Those are good examples. A purist might object (and I am sure has done so
on this mailing list) that factoring a rational such as 3/40 is silly since
it is a unit in the field QQ, but we make a very useful (for everyday work
by experts and for beginners) special case. In a sense, QQ knows that it
is the field of fractions of a unique factorization domain ZZ so that if
you ask for something like a factorization or a gcd *or a primality test*
then *of course* you mean that with respect to the parent ZZ not QQ.
Another place where useful "abuse notation" like this is in ideals of
number fields, which include all nonzero fractional ideals.
However pedantic you are it is very hard indeed to justify this for a
package which is intended for a wide class of users:
sage: a = 300/100
sage: a
3
sage: a in ZZ
True
sage: a.is_prime()
False
!
John
>
>
> Vincent
>
>
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