Question

Please fast :

p and q are primes

p+q=(p-q)^3

Find p and q

EXPLAIN your answer .

Answer 1

Given :

p + q = ( p - q )³

Let p - q be k

So : k + 2 q = p - q + 2 q = p + q

Thus : k³ = k + 2 q

==> k³ - k = 2 q

==> k ( k² -  1 ) = 2 q

==> k ( k + 1 )( k - 1 ) = 2 q

Now both 2 and q are primes !

So , they have only three factors 2 , q , 1

The least number is 1

The least number in the left is k - 1

So :

k - 1 = 1

==>k = 2

Hence p - q = 2 ....................(1)

Now :

( p - q )³ = p + q

==> p + q = 2³

==> p + q = 8 .....................(2)

Now add (1) and (2) to get :

2 p = 8 + 2

==> 2 p = 10

==> p = 10/2

==> p = 5

p - q = 2

==> 5 - q = 2

==> q = 5 - 2

==> q = 3

ANSWER :

The two primes are 5 and 3

Hope it helps :-)

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Answer 2

A/Q....

p + q = ( p - q )³

now....Let p - q be k

So

k + 2 q = p - q + 2 q = p + q

Thus

k³ = k + 2 q

=> k³ - k = 2 q

=> k ( k² -  1 ) = 2 q

=> k ( k + 1 )( k - 1 ) = 2 q

Now both 2 and q are primes

So , they have only three factors
.
2 , q , 1

The least number is 1

The least number in the left is( k - 1)

So

k - 1 = 1

=>k = 2

Hence p - q = 2 ....................(1)

Now

( p - q )³ = p + q

=> p + q = 2³

=> p + q = 8 .....................(2)

Now add (1) and (2) to get :

2 p = 8 + 2

=> 2 p = 10

=> p = 10/2

=> p = 5

p - q = 2

=> 5 - q = 2

=> q = 5 - 2

=> q = 3


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