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KingGeorge
 4 years ago
[UNSOVED] What is the largest number (in terms of a, b, c) that can not be expressed in the form \[ax+by+cz\]
where \[x,y,z≥0\]\[gcd(a,b,c)=1\]and all values are nonnegative integers?
As an example, take \[6x+10y+15z\]Here, the largest integer that can't be expressed in this form given \[x, y, z \geq 0\]is 29.
If instead you're solving the same problem, only with \[ax+by\]instead of\[ax+by+cz\] with the same conditions that every value is a nonnegative integer, the largest integer not expressible in this form is given by the formula\[abab\]
KingGeorge
 4 years ago
[UNSOVED] What is the largest number (in terms of a, b, c) that can not be expressed in the form \[ax+by+cz\] where \[x,y,z≥0\]\[gcd(a,b,c)=1\]and all values are nonnegative integers? As an example, take \[6x+10y+15z\]Here, the largest integer that can't be expressed in this form given \[x, y, z \geq 0\]is 29. If instead you're solving the same problem, only with \[ax+by\]instead of\[ax+by+cz\] with the same conditions that every value is a nonnegative integer, the largest integer not expressible in this form is given by the formula\[abab\]

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anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0A similar problem appear in IMO 1983, Let \( a, b,\) and \(c\) be positive integers, no two of which have a common divisor greater than \(1\). Show that \(2abc − ab − bc − ca\) is the largest integer that cannot be expressed in the form \(xab+yca+zab\), where\( x, y,\) and \(z\) are nonnegative integers.

KingGeorge
 4 years ago
Best ResponseYou've already chosen the best response.2no, the very last part of a homework problem from our textbook. I've figured out the two variable case without too much trouble, but for some reason, I just don't know how to generalize it.

KingGeorge
 4 years ago
Best ResponseYou've already chosen the best response.2Well, that appears to be almost the generalization I'm looking for...

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Yes, I think we still need to do some work, btw what level of mathematics is this?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Like highschool , undergrad, etc...

KingGeorge
 4 years ago
Best ResponseYou've already chosen the best response.2undergrad  intro to number theory

KingGeorge
 4 years ago
Best ResponseYou've already chosen the best response.2I've worked through a few other cases while varying c. So far I've gotten 25 if a=6, b=10, c=11. And 23 if a=6, b=10, c=9.

KingGeorge
 4 years ago
Best ResponseYou've already chosen the best response.2Here a few more cases 11 if (a, b, c) = (4, 5, 6) also 11 if (a, b, c) = (4, 5, 7) also 11 if (a, b, c) = (4, 6, 9)

KingGeorge
 4 years ago
Best ResponseYou've already chosen the best response.2My best guess currently, for those of you looking at this, is that the solution is the smallest integer that satisfies these equations\[\begin{matrix} ababd = cx \\ acacd=by \\ bcbcd=az \end{matrix}\]where d is the solution we're looking for, and x, y, z are still all positive.
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