What happened to elegance

Number Theory Level 4

find the smallest positive integer x such that ( x + y ) ( x − y ) = 1000001 (x+y)(x-y)=1000001 where x and y are positive integers


The answer is 5001.

Jerry Bao by Jerry Bao , 20, Canada

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1 solution

Kay Xspre
Sep 27, 2015

1000001 = 1 0 6 + 1 = ( 1 0 2 + 1 ) ( 1 0 4 − 1 0 2 + 1 ) = 101 × 9901 1000001 = 10^6+1 = (10^2+1)(10^4-10^2+1) = 101 \times 9901

Provided that both 101 and 9901 prime number and x + y > x − y x+y > x-y , there will be case where ( x + y , x − y ) = ( 9901 , 101 ) , ( 1000001 , 1 ) (x+y, x-y) = (9901, 101), (1000001, 1) . To find the minimum x x , we will choose the lowest value of ( x + y ) + ( x − y ) (x+y)+(x-y) , which is ( x + y , x − y ) = ( 9901 , 101 ) (x+y, x-y) = (9901, 101) . Solve then x = 5001 x = 5001

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Do you have a particular method for knowing that 9901 is prime ?

Louis LeLouis - 5 years, 8 months ago

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Smallest way is dividing it by prime numbers starting from 2 till the nearest prime to the floor function of its square root. If it divides no number in this list then its prime.

Kushagra Sahni - 5 years, 8 months ago

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