Opened 9 years ago

Last modified 9 years ago

#13516 closed defect

prime_powers doesn't work with start very well — at Version 14

Reported by: kcrisman Owned by: was
Priority: major Milestone: sage-5.6
Component: number theory Keywords: beginner
Cc: was Merged in:
Authors: Kevin Halasz Reviewers:
Report Upstream: N/A Work issues:
Branch: Commit:
Dependencies: Stopgaps:

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Description (last modified by dimpase)

See this sage-support thread.

In Sage 5.3, the function prime_powers behaves a little strange:

sage: prime_powers(4,10)
[4, 5, 7, 8, 9]
# As expected

sage: prime_powers(5,10)
[7, 8, 9]
# 5 isn't a prime power anymore???

# And now things become even worse:
sage: prime_powers(7,10)
---------------------------------------------------------------------------
IndexError                                Traceback (most recent call last)

/home/mueller/<ipython console> in <module>()

/home/mueller/local/sage-5.3/local/lib/python2.7/site-packages/sage/rings/arith.pyc in prime_powers(start, stop)
    743     i = bisect(v, start)
    744     if start > 2:
--> 745         if v[i] == start:
    746             i -= 1
    747         w = list(v[i:])

IndexError: list index out of range

Yeah, this seems problematic. The code in question is old, too, so perhaps there is a more efficient way to do it in the meantime...

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  1. patch

Change History (14)

comment:1 Changed 9 years ago by kcrisman

  • Keywords beginner added

comment:2 Changed 9 years ago by dimpase

as prime_powers(m) works, an easy workaround is to re-implement prime_powers(m,n) as the difference of prime_powers(n) and prime_powers(m-1).

comment:3 Changed 9 years ago by khalasz

  • Description modified (diff)
  • Status changed from new to needs_review

comment:4 follow-up: Changed 9 years ago by khalasz

I found the code to be riddled with errors, so I decided to completely rework it. Also, I have qualms about calling 1 a prime power, but did so because the old function did. If you think its fine to drop this, let me know.

comment:5 in reply to: ↑ 4 Changed 9 years ago by kcrisman

Thanks for your work - hopefully someone will review it soon. You can put your real name in the author area.

Also, I have qualms about calling 1 a prime power, but did so because the old function did. If you think its fine to drop this, let me know.

Well, John Horton Conway does call -1 a prime, in which case every nonzero integer (not just positive) is a unique product of prime powers - not a unique product of primes, note, nor of the exponents, but of the prime powers themselves (I can't find a link for this right now, my apologies) in which case positives get the power 1 and and negatives -1. I think that's right... anyway, maybe they were thinking this?

comment:6 Changed 9 years ago by was

I would prefer to leave 1 as a prime power because it is listed in Sloane's tables as a prime power: http://oeis.org/A000961

There he says "Since 1 = p0 does not have a well defined prime base p, it is sometimes not regarded as a prime power.", which might be where your misgivings come from.

If by "prime power" one thinks of "power of a prime", the only question is in what set are we considering the prime powers. If we take the natural numbers, then the number 1 is definitely a power of a prime.

If by "prime power" one thinks "power of a specific canonical prime", then 1 is not such a thing.

In this case, the best thing to do is stick with what is there (to avoid introducing bugs in other people's code!) and clearly document/define what a prime power is in Sage.

comment:7 Changed 9 years ago by khalasz

  • Authors set to Kevin Halasz

comment:8 Changed 9 years ago by khalasz

I just updated the docstring to make the fact that 1 is a prime power explicit.

comment:9 Changed 9 years ago by kcrisman

Could you speed this up slightly by making s = stop.sqrt() or something so that it's not computed for each prime. In fact, even that is a more expensive comparison each time because stop.sqrt() is likely a symbolic element, so maybe even stop.sqrt().n() would be appropriate... Also, once p >stop.sqrt(), presumably all remaining p are beyond it as well, so maybe there could be some speedup there too. Just some ideas.

comment:10 Changed 9 years ago by khalasz

I've changed the patch so that s=stop.sqrt() is calculated outside of the for loop. After some tests, I saw that this was faster than setting s=stop.sqrt().n().

Also, note that when p>s, the content of that if loop is a break command, meaning that the entire for loop ends. Thus, once a single p>s, no more p values are tried.

comment:11 Changed 9 years ago by dimpase

  • Status changed from needs_review to needs_work

The comment on line 708 in sage/rings/arith.py needs to be fixed, too - it talks about primes rather than prime powers.

I also think that the following:

       sage: prime_powers(10,7) 
 	761	        Traceback (most recent call last): 
 	762	        ... 
 	763	        ValueError: the first input must be less than the second input, however, 10 > 7 

i.e. the corresponding implementation logic is not right, in the sense that it should just return empty lists rather than throwing exceptions. And negative start should be allowed too (cf. the semantics of range()).

Then, in the following fragment

 	783	    output = prime_range(start,stop) 
 	784	    if start == 1: 
 	785	        output.append(1) 
 	786	     
 	787	    s = stop.sqrt() 
 	788	    for p in prime_range(stop): 

prime_range(), which is not cheap, is basically called two times instead of one. One can do with one call to prime_range(stop) just fine.

comment:12 Changed 9 years ago by khalasz

  • Status changed from needs_work to needs_review

Dimpase, I've addressed all of your suggestions. Let me know what you think of these changes/if you have other suggestions.

comment:13 Changed 9 years ago by dimpase

  • Description modified (diff)

comment:14 Changed 9 years ago by dimpase

  • Description modified (diff)
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