A equation involving prime.
Find sum of all positive integers , and primes such that
.
For example, if the solutions are and then the answer will be
This is a INMO problem.
The answer is 23.
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1 solution
f 1 = m ( 4 m 2 + m + 1 2 ) − 3 ( p n − 1 ) m ∣ x , 1 , 5 0 p n I l e t p v a r y f r o m 5 , 7 , 1 1 , 1 3 , 1 7 a n d n = 1 t o 7 . x w a s a d j u s t e d s o i n t h e l i s t f o r f t h e r e w e r e b o t h + t i v e a n d − t i v e v a l u e s . E x c e p t f o r ( 1 2 , 4 , 7 ) , f j u m p e d f r o m + t i v e t o − t i v e v a l u e s . B e l o w t h e r e l e v a n t o b s e r v a t i o n s ( m , n , p ) . . . . . . . . . . f m = . . . . . . + t i v e ( m + 1 , n , p ) . . . . . f m + 1 = . . . . − t i v e ( 1 , 1 , 7 ) . . . . . f 1 = 1 , ( 2 , 2 , 7 ) . . . . . f 2 = 8 4 , ( 6 , 3 , 7 ) . . . . . f 6 = 5 4 ( 2 , 1 , 7 ) . . . . . f 2 = − 4 2 , ( 3 , 2 , 7 ) . . . . . f 3 = − 9 , ( 7 , 3 , 7 ) . . . . . f 7 = − 4 7 9 , ( 1 1 , 4 , 7 ) . . . . . f 1 1 = 1 6 2 3 ( 1 2 , 4 , 7 ) . . . . . f 1 2 = 0 , ( 1 3 , 4 , 7 ) . . . . . f 1 3 = − 1 9 1 3 , ( 2 3 , 5 , 7 ) . . . . . f 2 3 = 9 4 5 , ( 1 , 1 , 1 1 ) . . . . . f 1 = 1 3 , ( 9 , 3 , 1 1 ) . . . . . f 9 = 8 8 5 , e c t . ( 2 4 , 5 , 7 ) . . . . . f 2 4 = − 5 7 4 2 , ( 2 , 1 , 1 1 ) . . . . . f 2 = − 3 0 , ( 1 0 , 3 , 1 1 ) . . . . . f 1 0 = − 2 3 0 , e c t . B e l o w i s t h r o u g h g r a p h . S w e e p i n g a n d c h e c k i n g b y m a g n i f y i n g t o s e e w h e r e x = w i s a n i n t e g e r .
