Простые числа в интервалах

Даётся элементарное доказательство постулата Бертрана.
Сформулированы более строгие утверждение о распределении простых чисел
в интервалах между натуральными числами.

THE  146th-147th PROBLEMS
ELEMENTARY PROOF OF BERTRAND’S POSTULATE
AL AFLITUN’S CONJECTURES
ON THE PRIME NUMBER DISTRIBUTION

        Lemma: Every prime number can be represented as a difference between two positive integers in   
        infinite ways, and there are two or more primes in same interval between these integers :
(1)   ;                p;_i=m_i1-n_i1=m_i2-n_i2 ;=;=m;_ij-n_ij=;;    
                {i,j}=1,2,3,…; ;{p_i  are primes}=2,3,5,…;m;_ij=3,4,5,…,n_ij=1,2,3,…;
                i.e.
        there exist definite inclusions of p_i in p_i intervals
(2)                p_i;(n_i1,m_i1 ),…,p_i;[n_(;ip;_i );p_i,m_(;ip;_i );;2p;_i ).

        From the last inclusion in (2) for the maximal inclusion interval we obtain
(3)                2n_(;ip;_i )= m_(;ip;_i ) ,
        that  proves  well known Bertrand’s postulate.

        Conclusion:Between 1 and;  2p;_i there exist p_i intervals in which there is the prime sequence 3,…, p_i.
       
        Examples:
        1) 2=3-1=4-2=…, 2;(1,3), 2;[2,4); 2) 3=4-1=5-2=6-3=…, 3;(1,4), 3;(2,5), 3;[3,6);
        3) 5=6-1=7-2=8-3=9-4=10-5=…, 5;(1,6), 5;(2,7), 5;(3,8), 5;(4,9), 5;[5,10);
        4) 7=8-1=9-2=10-3=11-4=12-5=13-6=14-7=…, 7;(1,8), 7;(2,9), 7;(3,10), 7;(4,11), 7;(5,12), 7;[6,13),
            7;[7,14).
THE  146th PROBLEM
Prove above conjecture.

OTHER CONJECTURES
ON THE PRIME NUMBER DISTRIBUTION IN INTERVALS

1.There are prime numbers distributed in intervals [m^2+1,m^2+m+1]  with
m=1,2,3,4,…, and there are at least two prime numbers in each interval.
As a special case of this conjecture we have the conjecture, proposed by Adrien-Marie Legendre: there is a prime number between m^2 and ;(m+1);^2   for every positive integer m.
2. There are all prime numbers distributed in intervals [2m-5,2m-3]  with
m=3,4,5,6,….
THE  147th PROBLEM
Prove above conjectures.