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![[parent]](http://images.planetmath.org:8080/images/uparrow.png) Viewing Correction to 'Bezout's lemma (number theory)'
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switch a,b and x,y; operatorname; name of entry by djao
Correction id: 548
Filed on: 2002-05-27 20:48:44
Status: Accepted on 2002-05-28 09:30:02
Type: Erratum
Correction text:
I recommend changing the text to the following, and renaming the entry to something like "solution of linear Diophantine equation", or something other than what it is now:
Let $a,b$ be integers, not both zero. Then there exist integers $x,y$ such that
$$
ax + by = \gcd(a,b).
$$
Description of changes, and rationale:
1) In almost every treatment I have seen, the letters a,b are used for constants and the letters x,y are used for variables. It would be good to maintain that convention here. It is at the least jarring to see the roles not only disregarded, but *reversed*.
2) Do not use "gcd" for the symbol gcd. "gcd" in math mode is interpreted as a product of the three variables g, c, and d. There is a built in command called \gcd which is much preferable for the present purpose. Even were it not for such a command, you should use \operatorname{gcd} instead of just plain gcd.
3) The quantifier "There exist" is standard. The quantifier "there are always", while perfectly valid english, is not in standard use in mathematics.
4) Use "not both zero" instead of "both not zero". The result does not require both $a,b$ to be nonzero, but only one.
5) The comma after the variables $x,y$ seems unnecessary in such a short phrase.
6) I have never heard this fact cited by the name "Bezout's Lemma", although in retrospect seems it is sometimes called Bezout's Lemma by others. In any case it is unfair to give primary credit to Bezout, since this result predates Bezout by millennia. For this reason I would relegate the name Bezout's Lemma to a synonym if you have to mention it at all. |
Comment from object owner mathwizard:
I recommend changing the text to the following, and renaming the entry to something like "solution of linear Diophantine equation", or something other than what it is now:
Let $a,b$ be integers, not both zero. Then there exist integers $x,y$ such that
$$
ax + by = \gcd(a,b).
$$
Description of changes, and rationale:
1) In almost every treatment I have seen, the letters a,b are used for constants and the letters x,y are used for variables. It would be good to maintain that convention here. It is at the least jarring to see the roles not only disregarded, but *reversed*.
2) Do not use "gcd" for the symbol gcd. "gcd" in math mode is interpreted as a product of the three variables g, c, and d. There is a built in command called \gcd which is much preferable for the present purpose. Even were it not for such a command, you should use \operatorname{gcd} instead of just plain gcd.
3) The quantifier "There exist" is standard. The quantifier "there are always", while perfectly valid english, is not in standard use in mathematics.
4) Use "not both zero" instead of "both not zero". The result does not require both $a,b$ to be nonzero, but only one.
5) The comma after the variables $x,y$ seems unnecessary in such a short phrase.
6) I have never heard this fact cited by the name "Bezout's Lemma", although in retrospect seems it is sometimes called Bezout's Lemma by others. In any case it is unfair to give primary credit to Bezout, since this result predates Bezout by millennia. For this reason I would relegate the name Bezout's Lemma to a synonym if you have to mention it at all. |
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