Hello

^{32  = 9,    42  = 16,   52  = 25}

          ^{9 + 16 = 25}

        ^{(5 - 4)*(5 + 4) = 1*9 = 25 - 16 = 9}

       ^{(5 - 3)*(5 + 3) = 2*8 - 25  - 9 = 16}

  ^{32  = 9 < 11 < 13 < 16 = 42} 

 ^{Hence the open interval ((32, 42 )) = ((9, 16)) contains two prime numbers 11 and 13.}

  ^{42  = 16 < 17 < 19 < 23 < 25 = 52} 

^{Hence the open interval ((42, 52 )) = ((16, 25)) contains three prime numbers 17, 19, and 23.}

  ^{52  = 25 < 29 < 31 < 37 < 41 < 43 < 47 < 49 = 72} 

^{Hence the open interval ((52, 72 )) = ((25, 49)) contains six prime numbers 29, 31, 37, 41, 43, and 47.}

Note that 6²  = 36.

      ^{52  = 25 < 29 < 31 < 37 < 41 < 43 < 47}

       ^{< 49 = 72 < 53 < 59 < 61 <  67 < 71 < 73  < 79 < 83 < 89 < 97 < 101 < 103 < 107 < 109 < 113} 

       ^{<  121 = 112  < 127 < 131 < 137 < 139 < 149 < 151 < 157 < 163}

       ^{<  169 = 132 < 173 < 179 < 181 < 191 < 193  <  197  < 199 < 211 < 223 < 227 < 229  < 233 < 239  < 241}

                           ^{<  251  < 257 < 263 < 269 < 271 < 277  <  281  < 283}

        ^{< 289 = 172  <   ...}

Let us denote  the set { x | a < x < b and x is a real nmber } by (( a, b )) which is called

an open interval with end points a and b.

^{Hence the open interval (( 52, 62 )) = (( 25,  36 )) contains two prime numbers 29 and 31.}

^{And the open interval (( 62, 72 )) = (( 36,  49 )) contains four prime numbers 37, 41, 43, and 47.}

Remember the Legendre's conjecture [1808]:

For each positive integer n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...

the open interval (( n²,  (n+1)² )) contains as least one prime number.

^{It is very easy to check that the open interval (( 12,  22 )) = (( 1, 4 )) contains two prime numbers 2 and 3.}

The following is my modified conjecture:

For each positive integer n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...

the open interval (( n²,  (n + 1)² )) = (( n*n,  (n+1)*(n + 1) )) contains as least two prime numbers.

How can prove that ?

Answer:   ??????????????????