# Can two different triads of natural numbers give the same results to these operations?

If I have a triad of natural numbers $$a$$, $$b$$ and $$c$$, and another triad $$a'$$, $$b'$$ and $$c'$$, such that either $$a\neq a'$$, $$b\neq b'$$, or $$c\neq c'$$, is it possible that these equations are true?

$$ab+c=a'b'+c'$$

$$ac+b=a'c'+b'$$

$$cb+a=c'b'+a'$$

If I had only the first equation, it is obviously possible, for example $$a=8$$, $$b=2$$ and $$c=14$$ give the same result as $$a'=10$$, $$b'=2$$ and $$c'=10$$, when we introduce the other two equations it feels like it would be impossible but I don't know how to prove it

There are (perhaps some might say trivial) counterexamples of the form $$(1,m,n)$$ and $$(1,n,m)$$--for instance $$a=1,b=3,c=4$$ and $$a'=1, b'=4, c'=3$$

There are also nontrivial counterexamples such as: $$a=2,b=5,c=89 {\rm \;\; and\;\; } a'=5,b'=13, c'=34$$

I found this with a search using Python.

I just noticed: They are all Fibonacci numbers.

Here's one that isn't all Fibonacci numbers: $$a=2,b=9,c=67 {\rm \;\; and\;\; } a'=3,b'=14, c'=43$$

• Thank you very much! The relationship with Fibonacci was very surprising, do you have any ideas of why it shows up? Jun 30, 2022 at 0:52
• I have no idea why the one with all Fibonacci number's showed up! Jun 30, 2022 at 0:53
• It's very interesting, thank you again! Jun 30, 2022 at 1:02