The Frobenius Coin Problem with a Twist on Gaussian Integers Let S be the set of Gaussian integers of the form a+bi, where a and b are non-negative integers, and a+b≤100. Consider two "coin denominations" z 1 ​ =3+2i and z 2 ​ =2+5i. A Gaussian integer N=x+yi (where x,y are non-negative integers) is said to be "representable" if there exist non-negative integers c 1 ​ and c 2 ​ such that N=c 1 ​ z 1 ​ +c 2 ​ z 2 ​ . Now, let U be the set of all unrepresentable Gaussian integers N=x+yi (with x,y≥0) such that N∈S. The Question: Find the largest "norm" of an unrepresentable Gaussian integer in S. The norm of a Gaussian integer a+bi is defined as N(a+bi)=a 2 +b 2 . Determine the number of unrepresentable Gaussian integers in S whose real part is equal to their imaginary part (i.e., x=y). Why this is hard: Gaussian Integers: It moves beyond standard integers into a new ring, requiring understanding of their arithmetic and properties. Frobenius Coin Problem Extension: The classic Frobenius Coin Problem deals with integers. This extends it to Gaussian integers, making "representability" more complex. The concept of "largest unrepresentable" (Frobenius number) is tricky here due to the 2D nature. Constraints (a+b≤100): This adds a boundary condition that limits the search space but also requires careful consideration of what's within the set S. "Norm" as a measure: Finding the largest unrepresentable based on its norm, rather than just its real or imaginary part, adds another layer of complexity to the optimization. Specific structure of z 1 ​ and z 2 ​ : Their real and imaginary parts are relatively prime, but their complex nature means standard GCD arguments don't directly apply in the same way. Part 2 - Specific Subset: Focusing on x=y further narrows the problem, requiring a different approach to enumeration.

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Welcome to an advanced exploration of the Frobenius coin problem extended to Gaussian integers. In the classic problem, we ask what is the largest amount that cannot be made using given coin denominations. Today, we work with complex coin denominations z1 equals 3 plus 2i and z2 equals 2 plus 5i in the complex plane. Our goal is to find which Gaussian integers cannot be represented as non-negative integer combinations of these complex coins. To determine which Gaussian integers are representable, we set up a system of linear equations. A Gaussian integer N equals x plus yi is representable if it can be written as c1 times 3 plus 2i plus c2 times 2 plus 5i, where c1 and c2 are non-negative integers. This gives us x equals 3c1 plus 2c2 and y equals 2c1 plus 5c2. Solving this system, we get c1 equals 5x minus 2y over 11 and c2 equals 3y minus 2x over 11. For representability, both c1 and c2 must be non-negative integers. Now we find the largest norm of unrepresentable Gaussian integers in set S. The norm of x plus yi is x squared plus y squared. Since we're constrained by x plus y less than or equal to 100, the maximum norm occurs on the boundary line x plus y equals 100. The function x squared plus y squared is maximized at the endpoints: 100 comma 0 and 0 comma 100, both giving norm 10000. We can verify both points are unrepresentable: for 100 comma 0, the condition 3y less than 2x fails, and for 0 comma 100, the condition 5x less than 2y fails. Therefore, the largest norm is 10000. Now we count unrepresentable Gaussian integers where x equals y. For N equals k plus ki, we need k plus k less than or equal to 100, so 0 less than or equal to k less than or equal to 50. The unrepresentability conditions simplify: 5k less than 2k and 3k less than 2k are both false for non-negative k. The only condition is k not congruent to 7k modulo 11, which simplifies to 6k not congruent to 0 modulo 11. This means k is unrepresentable if and only if k is not a multiple of 11. In the range 0 to 50, there are 51 total values and 5 multiples of 11: 0, 11, 22, 33, and 44. Therefore, 51 minus 5 equals 46 unrepresentable points. We have successfully solved the Frobenius coin problem extended to Gaussian integers. Using coin denominations z1 equals 3 plus 2i and z2 equals 2 plus 5i, we found that the largest norm of an unrepresentable Gaussian integer in set S is 10000, achieved by the points 100 plus 0i and 0 plus 100i. For the special case where x equals y, we counted 46 unrepresentable Gaussian integers out of 51 total possibilities. This elegant extension of the classical Frobenius problem demonstrates the rich mathematical structure that emerges when we move from integers to Gaussian integers, opening new avenues for exploration in algebraic number theory.