Vocabulary/qco

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q: y	Prime Factors

Rank 0 -- operates on individual atoms of y, producing a result that may require fill -- WHY IS THIS IMPORTANT?

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The prime factorization of integer y, listed in ascending order.

The same as (3 p: y). See Primes (p:).

   q: 2^31
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
   q: 1+ 2^31
3 715827883
   q: 2+ 2^31
2 5 5 13 41 61 1321

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Common uses

Mathematical investigations.

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Related Primitives

Primes (p:)

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x q: y

Prime Exponents

Rank 0 0 -- operates on individual atoms of x and y, producing a result that may require fill -- WHY IS THIS IMPORTANT?

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Selected primes and their exponents in the factorization of y.

An exponent of 0 means that the corresponding prime is not a factor of y.

   p: i.10   NB. the first 10 primes
2 3 5 7 11 13 17 19 23 29
   q: 700    NB. prime factors of 700
2 2 5 5 7
   _ q: 700   NB. 700 = (2^2) * (3^0) * (5^2) * (7^1)
2 0 2 1
   __ q: 700
2 5 7
2 2 1

The rank of the result, and the values listed, depend on x.

x	Result Rank	Meaning of Results	Number of Primes	Zero exponents included?
positive	1	The leading exponents of the prime factorization (keeping primes in ascending order)	x (x=_ means "up through the last nonzero exponent")	yes
negative	2	Second row: The trailing exponents of the prime factorization (keeping primes in ascending order)	|x (x=__ means "all nonzero exponents")	no
		First row: the corresponding primes

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Common uses

1. Calculate the number of divisors of a number

   */ >: _ q: 17
2
   */ >: _ q: 60
12

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More Information

1. When x is negative:

• the table t =. primes,:exponents is calculated, including primes with 0 exponents, ending with the largest prime with a nonzero exponent;
• The last (|x)<.({:$t) columns of t are selected (i. e. the last |x columns, but no more columns than exist in t);
• Columns with 0 exponents are then deleted.

Because the zero exponents are deleted at the last step, the result of  x q: y is not the same as that of  x {. __ q: y .

   _3 q: 2*3*5*17
17
 1

2.  __ q: y is equivalent to  2 p: y .

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Related Primitives

Primes (p:)

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Details

1. The result has integer or extended-integer precision, depending on the size of the largest prime factor.

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