Set with an Interesting Property...

Number Theory Level 2

The set $\{1,3,8,120\}$ has the interesting property that if you take any two numbers in that set and multiply them together, you will get precisely one less than a square number.

And one more number to this set such that it retains its property. (Your answer will be the number you add to the set.)

Bonus: Is than more than one number that you can add to this set such that it retains its property?

The answer is 0.

2 solutions

A Former Brilliant Member
Nov 9, 2018

No such fifth positive integer is possible

Mr. Mendrin, I presume. :)

Brian Charlesworth - 2 years, 7 months ago

only from the Y!A days... I'm starting to miss them

A Former Brilliant Member - 2 years, 7 months ago

Yeah, I know what you mean.

Brian Charlesworth - 2 years, 7 months ago

@Brian Charlesworth – @Brian Charlesworth , sir , does Mr.Mendrin have any other 'active account'? These have been deleted and his problems are also nice

Mr. India - 2 years, 3 months ago

@Mr. India – Yes, you can enter Michael Mendrin into the search field to gain access to his problems, or go here to see his profile page.

Brian Charlesworth - 2 years, 3 months ago

@Brian Charlesworth – Ok thank you, has he stopped using brilliant?

Mr. India - 2 years, 3 months ago

@Mr. India – He is not as active as he used to be, but I don't think he has completely stopped.

Brian Charlesworth - 2 years, 3 months ago

Proof? $\quad$

Pi Han Goh - 2 years, 7 months ago

Ask Mendrin for the proof. He knows everything. He'll probably say, "go look up this one" D(-1) Quadruple Conjecture or something like that

A Former Brilliant Member - 2 years, 7 months ago

Mendrin? Never heard of him. I know someone who is omniscience though, he goes by the name Scythian1950.

Pi Han Goh - 2 years, 7 months ago

You could ask Mendrin, of course. But Baker and Davenport proved this back in 1969; their paper is behind a paywall, but this paper goes further and provides a generalization of their result.

@Michael Scythian Hahaha I posted my comment at the same time you were adding the reference to yours. At least they are different papers, so Pi Han Goh has a choice. :)

Brian Charlesworth - 2 years, 7 months ago

Let me go check out Baker & Davenport.

Problem of Diophantus

says that the Diophantine pair {1, 3} cannot be extended to a Diophantine qunituple, so I think this paper is definitive on this matter, whether or not there's a 5th term {1, 3, 8, 120, ?}. Ask Mendrin for details on this.

A Former Brilliant Member - 2 years, 7 months ago

Joshua Lowrance
Nov 9, 2018

If you multiply any number in that set by $0$ , it will be $0$ , which is one less than $1^{2}$ .

As Michael points out, (and I'm guessing you already know, otherwise there wouldn't have been a unique solution), there is no fifth positive integer, as noted here .

Brian Charlesworth - 2 years, 7 months ago

I was not aware there was no fifth positive integer, I was just aware that if there was one, it was incredibly big, or it hasn't been discovered yet. Thank you for letting me know.

Joshua Lowrance - 2 years, 7 months ago

Set with an Interesting Property...

The answer is 0.

2 solutions

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