Solve for a,b,c,d: Hints & Tips

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The discussion revolves around finding integer solutions for the equations given specific constraints. Participants explore various methods, including programming in Excel and Haskell, to identify potential solutions. Four valid solutions within the range of 1 to 20 are identified: [1,1,2,6], [1,2,2,6], [2,2,2,5], and [2,2,3,3]. The conversation highlights the growth rates of the expressions involved to argue that these may be the only solutions. The thread emphasizes the use of coding for efficient solution generation and mathematical reasoning to validate findings.
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Can anyone give me hints on the following question.I do not know how to proceed.

Find all integers a,b,c,d satisfying the following relations
i) 1 \leq a \leq b \leq c \leq d
ii) ab+cd = a+b+c+d+3

thanks
 
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Well, since 1,2,3,4 didn't work, I'd try next to write a quick program or use Excel to see what some of the solutions look like.
 
Well I got at least one solution in Excel. Now if I could prove that it's the only one...
 
I got 4 solutions using Haskell
[[1,1,2,6],[1,2,2,6],[2,2,2,5],[2,2,3,3]]
These are the only solutions where all values are between 1 and 20. To show that they are they only four (which I'd guess they are) you can use an argument based on how fast ab + cd grows versus how fast a + b + c + d + 3 grows.

Incidentally, because I like to show off Haskell, this is what my code looks like
Code:
-- f just generates all possible lists of length k where each element is at least as great as the next element.
-- I wanted to do this efficient-like, which is why this may be a little confusing.  
f min n 0 = [[]]
f min n k = foldr (++) [] [[a:as | as <- (f a n (k-1))] | a <- [min..n]]

-- If I had done it the easy way instead of the efficient way using f 
-- then then I would have just let x = [[a,b,c,d] | a<-[1..20],b<-[1..20],c<-[1..20],a*b+c*d==a+b+c+d+3 && a >= b && b >= c && c >= d]
-- and not defined f or y
y = f 1 20 4
x = [[a,b,c,d] | [a,b,c,d] <- y, a*b+c*d==a+b+c+d+3]
Then in the interpreter I just typed x.
 
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write the above equation like (a-1)(b-1) + (c-1)(d-1) = 5... also, because of the first condition the second term is greater than or equal to the first term...and since they are all positive integers.....

can you work out the rest...?
 
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