About connection of one matrix of composite numbers with Legendre’s conjecture Garipov Ilshat Ilsurovich Russia, Republic of Tatarstan, Naberezhnye Chelny e-mail: [email protected]. Abstract Scientific paper is devoted to establish connection of T -matrix – matrix of composite numbers 6h ± 1 in special view – with Legendre’s conjecture. Keywords: T -matrix, prime numbers, leading element, upper and lower defining elements of number, ¾active¿ set for numbers (m − 1)4 and m4, ¾critical¿ element for numbers (m − 1)4 and m4, Legendre’s conjecture, ¾Weak¿ and ¾Strong¿ conjectures. List of symbols N0 – set of all natural numbers with zero. N – set of all natural numbers. Z – set of all integers. P – set of all prime numbers. R – set of all real numbers. T – matrix comprising all defining and all not defining elements. Te – set of all elements of T-matrix. D(b) – T -matrix upper defining element of number b. d(b) – T -matrix lower defining element of number b. Dk(b) – upper defining element of number b in k-row (k > 1) of T -matrix. dk(b) – lower defining element of number b in k-row (k > 1) of T -matrix. W (b) – T -matrix upper element of number b. arXiv:2104.06261v1 [math.GM] 7 Apr 2021 w(b) – T -matrix lower element of number b. Wk(b) – upper element of number b in k-row (k > 1) of T -matrix. wk(b) – lower element of number b in k-row (k > 1) of T -matrix. DT – set of all defining elements of T -matrix. nDT – set of all not defining elements of T -matrix. MT – set of all leading elements of T -matrix. DTk – set of all defining elements in k-row (k > 1) of T -matrix. π(x) – function counting the number of prime numbers less than or equal to x 2 R. 1 πMT (x) – function counting the number of T -matrix leading elements less than or equal to x 2 R. #k(a) – number of element a in k-row of T -matrix. 4 4 H(m−1)4; m4 – ¾active¿ set for numbers (m − 1) ; m . 4 4 C(m−1)4; m4 – ¾critical¿ element for numbers (m − 1) ; m . νk(x) – function counting the number of elements, less than or equal to x 2 R, in k-row of T -matrix. ν(x) – function counting the number of naturals of the form 6h ± 1, less than or equal to x 2 R. a%b – remainder after dividing a 2 N by b 2 N. 2 2 qm – number of prime numbers between m and (m + 1) . Introduction 1. T -matrix We construct a matrix T ≡ (a(k; n))∞×∞, where a(k; n) is a T -matrix element located in k-th row, n-th column and defined as follows: jnk n − 1 a(k; n) ≡ p(k) · 5 + 2 · + 4 · ; 2 2 1 where p(k) is the k-th element of sequence (p(k))k=1 of prime numbers: p(k) ≡ pk+2; (1) where pi is the i-th prime number in sequence of all prime numbers (see [1]). 1 Let (f(n))n=1 is a numerical sequence, where a common member f(n) is defined as follows: 3 − (−1)n f(n) ≡ 3n + : 2 THEOREM 1.1. (8k;n 2 N)(a(k; n) = p(k) · f(n)) : (2) DEFINITION 1.1. An element a(k; n) of matrix T is called defining if 1) a(k; n) is not divisible by 5; 2) a(k; n) can be expressed as a product of some two prime numbers, that is 5 6 ja(k; n) ^ (9p1;p2 2 P)(a(k; n) = p1 · p2): (3) DEFINITION 1.2. An element a(k; n) of matrix T is called not defining if he does not satisfy condition (3). DEFINITION 1.3. An element a(k; n) of matrix T is called leading if a(k; n) = p2(k): 2 DEFINITION 1.4. A T -matrix is called matrix comprising all defining and not defining elements. 1 LEMMA 1.2. (f(n))n=1 is a sequence of all numbers of the form 6h ± 1: 5; 7; 11; 13; 17; 19; 23; 25; ::: ; 6h − 1; 6h + 1; :::: 2 1 PROPERTY 1.1. The sequence (p (k))k=1 of T -matrix leading elements is ascending. The simplest properties and basic theorems about elements of T -matrix are proved in [1]. 2. About a T -matrix upper defining element of real number DEFINITION 2.1. A T -matrix defining element D(b) is called an upper defining element of number b 2 R : b > 49, if D(b) = min a(k1; n); a(k1;n)2DT a(k1;n)>b n2N where k1 is defined by condition 2 2 p (k1) = max p (k): 2 p (k)6b k>1 DEFINITION 2.2. A T -matrix defining element d(b) is called a lower defining element of number b 2 R : b > 49, if d(b) = max a(k2; n); a(k2;n)2DT a(k2;n)<b n2N where k2 is defined by condition 2 2 p (k2) = max p (k): p2(k)<b k>1 DEFINITION 2.3. A T -matrix defining element Dk(b) is called an upper defining element 2 of number b 2 R : p (k) 6 b, in k-row (k > 1) of T -matrix if Dk(b) = min a(k; n): a(k;n)2DT a(k;n)>b n2N DEFINITION 2.4. A T -matrix defining element dk(b) is called a lower defining element of number b 2 R : p2(k) < b, in k-row (k > 1) of T -matrix if dk(b) = max a(k; n): a(k;n)2DT a(k;n)<b n2N DEFINITION 2.5. A T -matrix element W (b) is called an upper element of number b 2 R : b > 49, if 3 W (b) = min a(k1; n); a(k1;n)2Te a(k1;n)>b n2N where k1 is defined by condition 2 2 p (k1) = max p (k): 2 p (k)6b k>1 DEFINITION 2.6. A T -matrix element w(b) is called a lower element of number b 2 R : b > 49, if w(b) = max a(k2; n); a(k2;n)2Te a(k2;n)<b n2N where k2 is defined by condition 2 2 p (k2) = max p (k): p2(k)<b k>1 DEFINITION 2.7. A T -matrix element Wk(b) is called an upper element of number b 2 R : 2 p (k) 6 b, in k-row (k > 1) of T -matrix if Wk(b) = min a(k; n): a(k;n)2Te a(k;n)>b n2N DEFINITION 2.8. A T -matrix element wk(b) is called a lower element of number b 2 R : p2(k) < b, in k-row (k > 1) of T -matrix if wk(b) = max a(k; n): a(k;n)2Te a(k;n)<b n2N LEMMA 2.1. (8k > 1) (8n > 1) (a(k; n) 2 DT , f(n) 2 Pnf2; 3; 5g) : (4) PROOF. Choose any k-row (k > 1) and any n-column (n > 1) of T -matrix. Necessity. Let a(k; n) 2 DT . By Definition 1.1, that means that 5 6 ja(k; n) ^ (9 p1; p2 2 P)(a(k; n) = p1 · p2): Then by rule (1), Lemma 1.2 and Theorem 1.1, p1 = p(k) 2 Pnf2; 3; 5g ^ p2 = f(n) 2 Pnf2; 3; 5g: It follows that f(n) 2 Pnf2; 3; 5g. The necessity is proved. 4 Sufficiency. Let f(n) 2 Pnf2; 3; 5g. It is clear that p(k) 2 Pnf2; 3; 5g for k > 1. p(k); f(n) 2 Pnf2; 3; 5g ) (2) ) p(k) · f(n) = a(k; n) ^ p(k); f(n) 2 P ^ 5 6 ja(k; n) ) (3) ) 5 6 ja(k; n) ^ (9 p1;p2 2 P)(a(k; n) = p1 · p2) , a(k; n) 2 DT : The sufficiency is proved. Lemma 2.1 is proved. THEOREM 2.2 (about the ¾transition down¿ of T -matrix defining element). 2 (8k; n 2 N) p (k) < a(k; n) ^ a(k; n) 2 DT ) 2 2 2 ) (9! j 2 N) k < j ^ a(k; n) < p (j) ^ a(j;#k(p (k))) = a(k; n) ^ a(j; n) = p (j) : (5) PROOF. Existence. It is established in [1]. Uniqueness. Suppose, 2 (8k; n 2 N)(p (k) < a(k; n) ^ a(k; n) 2 DT ) 2 2 ) (9 j1; j2 2 N)(j1 6= j2 ^ k < j1 ^ k < j2 ^ a(k; n) < p (j1) ^ a(k; n) < p (j2) ^ 2 2 ^ a(j1;#k(p (k))) = a(k; n) ^ a(j2;#k(p (k))) = a(k; n) ^ 2 2 ^ a(j1; n) = p (j1) ^ a(j2; n) = p (j2))): 2 2 a(j1;#k(p (k))) = a(k; n) ^ a(j2;#k(p (k))) = a(k; n) ^ j1 6= j2 ) 2 2 (2) ) a(j1;#k(p (k))) = a(j2;#k(p (k))) ^ j1 6= j2 , 2 2 , p(j1) · f(#k(p (k))) = p(j2) · f(#k(p (k))) ^ j1 6= j2 , , p(j1) = p(j2) ^ j1 6= j2 , j1 = j2 ^ j1 6= j2: As a result, a contradiction. The uniqueness is established. Theorem 2.2 is proved. COROLLARY 2.3. Let a(k; n), a(j; n) are T -matrix elements from Theorem 2.2. Then, a(k; n) a(k; n) − p(k) = p(j) − : p(k) p(j) PROOF. From algorithm №1 in [1], we get a(k; n) (9h 2 ) p2(k) + 2h · p(k) = a(k; n) , (9h 2 ) p(k) + 2h = : N N p(k) 5 2 2 a(k; n) (5) a(j;# (p (k))) (2) p(j) · f(# (p (k))) 2h = − p(k) = k − p(k) = k − p(k) = p(k) p(k) p(k) 2 p(j) · p(k) (2) a(j;# (p (k))) (5) a(k; n) = − f(# (p2(k))) = p(j) − k = p(j) − : p(k) k p(j) p(j) Corollary 2.3 is proved.