we defined the star operation on Z by a b ab a b for al

we defined the star operation * on Z by a * b = ab + a + b for all integers a and b.

Rewrite the following two statements as careful, precise mathematical statements and then prove them.

(a) The star of an integer with itself is always one less than a perfect square.

(b) The positive integers that cannot be written as stars of two positive integers are “the primes minus one.”

FOR (a) i do have the prove which is a*a = a² + 2a which is one less than perfect sequare but i\'m having difficult time to re-write the statement?

please show your work!

a> a*b = ab + a + b

if a = b

so the star of a is a* . a* represents the complex conjugate of a with a

a*a = a^2 +2a

adding 1 and subtraction 1 on the right hand side

a*a = a^2 + 2a +1 -1

= (a+1)^2 - 1

now we know that (a+1)^2 is a perfect square as it could be broken down to = (a+1)(a+1)

and this is the defination of perfect squares a number multiplied by itself

hence a*a = (a+1)^2 - 1 = perfect equare - 1 , a is an integer

hence proved