True or False :
All integers can be written as the sum of cubes of five integers, not necessarily distinct.
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It is true that all integers can be expressed as the sum of cubes of five integers, not necessarily distinct.
Note the identity 6 k = ( k + 1 ) 3 + ( k − 1 ) 3 − k 3 − k 3 We use the above identity for k = 6 n 3 − n = 6 n ( n − 1 ) ( n + 1 ) which is an integer for all n as the product of 3 consecutive numbers is divisible by 3 ! = 6 .
This gives us n 3 − n = ( 6 n 3 − n + 1 ) 3 + ( 6 n 3 − n − 1 ) 3 − ( 6 n 3 − n ) 3 − ( 6 n 3 − n ) 3
Therefore, n = ( − n ) 3 + ( 6 n − n 3 − 1 ) 3 + ( 6 n − n 3 + 1 ) 3 + ( 6 n 3 − n ) 3 + ( 6 n 3 − n ) 3 .