{"product_id":"extending-the-linear-diophantine-problem-a-k-a-the-frob-prob-von-curtis-kifer","title":"Extending The Linear Diophantine Problem","description":"\u003cp\u003eGiven integer-valued relatively prime `coins' a1; a2; :::; ak, the Frobenius number is the largest integer n such that the linear diophantine equation a1m1 + a2m2 + ::: + akmk = n has no solution in non-negative integers m1;m2; :::;mk. We denote by g(a1; :::; ak) the largest integer value not attainable by this coin system. That is to say that any integer x greater than the Frobenius number g(a1; :::; ak) has a representation x = a1x1 + a2x2 + ::: + akxk by a1; a2; :::; ak for some non-negative integers x1; x2; :::; xk. We say x is representable by a1; a2; :::; ak. While it is obvious that there are representable positive integers and non-representable positive integers, must there be a largest non-representable integer? Maybe there are indefinitely large non-representable integers for a1; a2; :::; ak with gcd (a1; a2; :::; ak) = 1. This notion of whether or not the Frobenius number is well-defined will be the first bit of mathematics we look at in this paper. Proposition 1.1. The Frobenius number g(a1; :::; ak) is well-defined. Proof. Given a1; a2; :::; ak with gcd (a1; a2; :::; ak) = 1, the extended Euclidean algorithm gives that there exist m1;m2; :::;mk 2 Z such that...\u003c\/p\u003e\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9783845405131\"\u003e\u003ch3\u003ea.k.a. 'The Frob prob'\u003c\/h3\u003e\u003c\/div\u003e","brand":"Libri","offers":[{"title":"Softcover - 9783845405131","offer_id":39442919686237,"sku":"9783845405131","price":49.0,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/d3775e38-77b5-4153-808a-c893a22b9a28.jpg?v=1773292993","url":"https:\/\/shop.autorenwelt.de\/products\/extending-the-linear-diophantine-problem-a-k-a-the-frob-prob-von-curtis-kifer","provider":"Autorenwelt Shop","version":"1.0","type":"link"}