perplexus.info :: Just Math : Inequality challenge

Inequality challenge (Posted on 2016-10-25)

Given two integers a and b such that:
I. a>b>1
ii. a+b divides ab+1
iii. a-b divides ab-1

Prove that a is less than b*sqrt(3).

No Solution Yet Submitted by Ady TZIDON

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re: Thoughts...hints toward formal proof

| Comment 2 of 4 |

(In reply to Thoughts by Brian Smith)

Hints toward formal proof:

Since ( i) (b^2-1)=(a+b)*b-(ab+1)

<ii> b^2-1 =(ab-1)-b(a-b)

(iii) b^2-1 must be divisible by (a+b)*(a-b) (iv)

.show that a and b must be relatively prime ( iv ) i

and a+b & a-b may have only 1 or 2 as common divider

SAY NO MORE.

Edited on November 17, 2016, 5:29 pm

Posted by Ady TZIDON on 2016-11-17 17:14:36

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