MHB Minimum Sum of Non-Negative Integers with Given Equation - POTW #504

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The problem involves finding non-negative integers a, b, c, and d such that the equation a² + b² - cd² = 2022 holds true, while minimizing the sum a + b + c + d. Initial guesses suggest values like a = b = 32, d = 1, and c = 26 yield a sum of 91, but other combinations, such as a = 45, b = 1, d = 2, and c = 1, result in a lower sum of 49. The discussion highlights the challenge of finding the optimal integers that satisfy the equation while minimizing the total sum. Participants express interest in discovering the correct solution to the problem. The search for the minimum sum continues as users explore various combinations of values.
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Here is this week's POTW:

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Let ##a,\,b,\,c## and ##d## be non-negative integers.

If ##a^2+b^2-cd^2=2022##, find the minimum of ##a+b+c+d##.

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Thanks for the interesting problem. I have not found the answer but I would like to guess it.
s:=a+b+c+d=a+b+d+\frac{a^2+b^2-2022}{d^2}
Guessing for minimum s that a=b and ##2a^2-2022## is least with an integer c
a=b=32, d=1, c=26
s=91
A nearby case is
a=33,b=31,d=1,c=28;\ s=93>91
However, for a>>b case
a=45,b=1,d=2,c=1;\ s=49
My guess failed. I will be glad to know the right answer.
 
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