Diophantine Equation P + L+ P + b ln +1 n+2 2n+1 Pn+1 = l1 ln P1 + L+ Pn + b Has Infinitely Many Prime Solutions
Chun-Xuan Jiang P. O. Box 3924, Beijing 100854, P. R. China liukxi @ public3. bta. net. cn
Abstract
By using the arithmetic function J 2n+1(w) we prove that Diophantine equation
P +L+ P +b ln +1 n+2 2n+1 Pn+1 = l1 l n P1 +L+ Pn + b
has infinitely many prime solutions.It is the Book proof. The J 2n+1(w) ushers in a new era in the prime numbers theory.
1
AMS Mathematics subject classification: Primary 11N05 11N32 11N35 11N36
A new branch of number theory: Santilli’s additive isoprime theory is introduced.
By using the arithmetic function J n (w) the following prime theorems have been proved [1-8]. It is the Book proof. 1. There exist infinitely many twin primes. 2. The Goldbach theorem. Every even number greater than 4 is the sum of two odd primes. 3. There exist finitely many Mersenne primes, that is, primes of the form 2 P -1 where P is prime. n 4. There exist finitely many Fermat primes, that is, primes of the form 2 2 +1. 5. There exist finitely many repunit primes whose digits (in base 10) are all ones. 6. There exist infinitely many primes of the forms: x 2 +1, x 4 +1, x8 +1, x16 +1. 7. There exist infinitely many primes of the form: x 2 + b.
8. There exist infinitely many prime m-chains, Pj+1 = mPj ± (m -1) , m = 2,3,L, including the Cunningham chains. 9. There exist infinitely many triplets of consecutive integers, each being the product of k distinct primes, (Here is an example: 1727913=3×11× 52361, 1727914=2×17×50821, 1727915=5×7×49369.) 10. Every integer m may be written in infinitely many ways in the form
P2 +1 m = k P1 -1
where k =1, 2, 3, L, P1 and P2 are primes. 11. There exist infinitely many Carmichael numbers, which are the product of three primes, four primes, and five primes. 12. There exist infinitely many prime chains in the arithmetic progressions. 13. In a table of prime numbers there exist infinitely many k-tuples of primes, where k = 2, 3, 4, L,105 .
2 14. Proof of Schinzel’s hypothesis.
15. Every large even number is representable in the form P1 + P2 LPn . It is the n primes theorem which has no almost-primes.
In this paper by using the arithmetic function J n (w) Diophantine equations are studied.
Theorem 1. Diophantine equation P +L+ P +b ln +1 n+2 2n+1 Pn+1 = , （1） l1 l n P1 +L+ Pn + b has infinitely many prime solutions, where b is the integer. We rewrite (1)
l1 l n ln +1 P2n+1 = (P1 +L+ Pn +b)Pn+1 - Pn+2 L- P2n - b . （2）
The arithmetic function [1-8] is 2n （ ） J 2n+1(w) = Õ ((P -1) - H(P)) ¹ 0, 3 3£P£Pi where w = Õ P is called the primorials. 2£P£Pi
Let H (P) denote the number of solutions of the congruence
l1 l n ln +1 (q1 +L+ qn + b)qn+1 - qn+2 L- q2n -b º 0 (mod P ), （4） where q j =1,2,L,P -1, j =1,2,L,2n.
Since J 2n+1 (w) ® ¥ as w ® ¥, there exist infinitely many primes P1,L,P2n such that P2n+1 is also a prime. It is the Book proof. It is a generalization of the Euler proof for the existence of infinitely many primes. The best asymptotic formula [1-8] is
p 2 (N,2n +1) = {P1, L, P2n : P1,L, P2n £ N; P2n+1 = prime }
3 J (w)w N 2n 2n+1 （ ） = 2n+1 2n+1 (1+ O(1)). 5 (2n)!(ln+1 + lmax )f (w) log N where lmax is a maximal value among (l1,L,ln ) , f(w) = Õ (P -1) is called the Euler function of the primorials. 2£P£Pi
Theorem 2. Diophantine equation
P3 + b P2 = （6） P1 + b has infinitely many prime solutions. The arithmetic function [1-8] is 2 （ ） J 3 (w) = Õ (P + 3P +3 - c(P))¹ 0 , 7 3£P£Pi where c(P) = -P + 2 if Pb; c(P) = 0 otherwise.
Since J 3 (w) ® ¥ as w ® ¥, there exist infinitely many primes P1 and P2 such that P3 is also a prime. The best asymptotic formula [1-8] is
p 2 (N,3) = {P1, P2 : P1,P2 £ N; P3 = prime }
J (w)w N 2 = 3 (1+ O(1)). （8） 4f 3 (w) log 3 N
Theorem 3. Diophantine equation P + b 3 （ ） P2 = 2 9 P1 + b has infinitely many prime solutions. The arithmetic function is 2 （ ） J 3 (w) = Õ (P - 3P + 3 + c(P)) ¹ 0 , 10 3£P£Pi
4 æ - bö where c(P) = P - 2 if Pb; c(P) = ç ÷ otherwise. è P ø
Since J 3 (w) ® ¥ as w ® ¥, there exist infinitely many primes P1 and P2 such that P3 is also a prime. The best asymptotic formula is
p 2 (N,3) = {P1, P2 : P1,P2 £ N; P3 = prime }
J (w)w N 2 = 3 (1+ O(1)). （11） 6f 3 (w) log 3 N
Theorem 4. Diophantine equation
2 P3 +1 P2 = （12） P1 +1 has infinitely many prime solutions. The arithmetic function is 2 （ ） J 3 (w) = Õ (P - 3P + 4) ¹ 0, 13 3£P£Pi
Since J 3 (w) ® ¥ as w ® ¥, there exist infinitely many primes P1 and P2 such that P3 is also a prime. The best asymptotic formula is
p 2 (N,3) = {P1, P2 : P1,P2 £ N; P3 = prime }
J (w)w N 2 = 3 (1+ O(1)). （14） 6f 3 (w) log 3 N
Theorem 5. Diophantine equation P +1 2 3 （ ） P2 = 2 , 15 P1 +1
5 has infinitely many prime solutions. The arithmetic function is 2 （ ） J 3 (w) = Õ (P - 3P + 3 + c(P))¹ 0 , 16 3£P£Pi where c(P) = 3 if 4 (P -1) ; c(P) = 1 otherwise.
Since J 3 (w) ® ¥ as w ® ¥, there exist infinitely many primes P1 and P2 such that P3 is also a prime. The best asymptotic formula is
p 2 (N,3) = {P1, P2 : P1,P2 £ N; P3 = prime }
J (w)w N 2 = 3 (1+ O(1)). （17） 8f 3(w) log 3 N
Theorem 6. Diophantine equation
P3 +b P2 = （18） P0 P1 + b
has infinitely many prime solutions, where P0 is an odd prime.
The arithmetic function is 2 （ ） J 3 (w) = Õ (P - 3P + 3 - c(P)) ¹ 0 , 19 3£P£Pi P-1
P0 where c(P) = -P + 2 if Pb; c(P) = -P0 +1 if b º1 (mod P ) ; P-1 P c(P) = 1 if b 0 º/ (mod P ); c(P) = 0 otherwise.
Since J 3 (w) ® ¥ as w ® ¥, there exist infinitely many primes P1 and P2 such that P3 is also a prime. The best asymptotic formula is
p 2 (N,3) = {P1, P2 : P1,P2 £ N; P3 = prime }
6 J (w)w N 2 3 （ ） = 3 3 (1+ O(1)). 20 2(P0 +1)f (w) log N
Theorem 7. Diophantine equation P +b P0 3 P2 = , （21） P1 + b has infinitely many prime solutions, where P0 is an odd prime. The arithmetic function is 2 （ ） J 3 (w) = Õ (P - 3P + 3 - c(P)) ¹ 0 , 22 3£P£Pi where c(P) = -P + 2 if Pb ; c(P) = -P0 +1 if P0 (P -1) ; c(P) = 0 otherwise.
Since J 3 (w) ® ¥ as w ® ¥, there exist infinitely many primes P1 and P2 such that P3 is also a prime. The best asymptotic formula is
p 2 (N,3) = {P1, P2 : P1,P2 £ N; P3 = prime }
J (w)w N 2 3 （ ） = 3 3 (1+ O(1)). 23 2(P0 +1)f (w) log N
Theorem 8. Diophantine equation P +1 2 3 （ ） P2 = 3 , 24 P1 +1 has infinitely many prime solutions The arithmetic function is 2 （ ） J 3 (w) = Õ (P - 3P + 3 - c(P)) ¹ 0 , 25 3£P£Pi where c(P) = -P + 8 if 3(P -1) ; c(P) = -1 otherwise.
Since J 3 (w) ® ¥ as w ® ¥, there exist infinitely many primes P1 and P2
7 such that P3 is also a prime. The best asymptotic formula is
p 2 (N,3) = {P1, P2 : P1,P2 £ N; P3 = prime }
J (w)w N 2 = 3 (1+O(1)). （26） 10f 3(w) log 3 N
Theorem 9. Diophantine equation
4 P3 +1 P2 = , （27） P1 +1 has infinitely many prime solutions The arithmetic function is 2 （ ） J 3 (w) = Õ (P - 3P + 3 - c(P)) ¹ 0 , 28 3£P£Pi where c(P) = -3 if 4 (P -1) ; c(P) = -1 otherwise.
Since J 3 (w) ® ¥ as w ® ¥, there exist infinitely many primes P1 and P2 such that P3 is also a prime. The best asymptotic formula is
p 2 (N,3) = {P1, P2 : P1,P2 £ N; P3 = prime }
J (w)w N 2 = 3 (1+O(1)). （29） 10f 3(w) log 3 N
Theorem 10. Diophantine equation P +1 3 （ ） P2 = 4 , 30 P1 +1 has infinitely many prime solutions. The arithmetic function is
8 2 （ ） J 3 (w) = Õ (P - 3P + 3 - c(P)) ¹ 0 , 31 3£P£Pi where c(P) = 1 if 4 (P -1) ; c(P) = -3 if 8(P -1) ; c(P) = 0 otherwise.
Since J 3 (w) ® ¥ as w ® ¥, there exist infinitely many primes P1 and P2 such that P3 is also a prime. The best asymptotic formula is
p 2 (N,3) = {P1, P2 : P1,P2 £ N; P3 = prime }
J (w)w N 2 = 3 (1+O(1)). （32） 10f 3(w) log 3 N
Theorem 11. Diophantine equation P +1 3 （ ） P2 = 6 , 33 P1 +1 has infinitely many prime solutions. The arithmetic function is 2 （ ） J 3 (w) = Õ (P - 3P + 3 - c(P)) ¹ 0 , 34 3£P£Pi
P-1 where c(P) = -5 if 12(P -1) ; c(P) = 1 if 6(P -1) ; c(P) = -(-1) 2 otherwise.
Since J 3 (w) ® ¥ as w ® ¥, there exist infinitely many primes P1 and P2 such that P3 is also a prime. The best asymptotic formula is
p 2 (N,3) = {P1, P2 : P1,P2 £ N; P3 = prime }
J (w)w N 2 = 3 (1+O(1)). （35） 14f 3(w) log 3 N
Theorem 12. Diophantine equation
9 P +1 3 （ ） P2 = 8 , 36 P1 +1 has infinitely many prime solutions. The arithmetic function is 2 （ ） J 3 (w) = Õ (P - 3P + 3 - c(P)) ¹ 0 , 37 3£P£Pi where c(P) = -7 if 16(P -1) ; c(P) = -3 if 8(P -1) ; c(P) = 1 if 4 (P -1) ; c(P) = 1 otherwise.
Since J 3 (w) ® ¥ as w ® ¥, there exist infinitely many primes P1 and P2 such that P3 is also a prime. The best asymptotic formula is
p 2 (N,3) = {P1, P2 : P1,P2 £ N; P3 = prime }
J (w)w N 2 = 3 (1+ O(1)). （38） 18f 3(w) log 3 N
Theorem 13. Diophantine equation
P4 + P5 +1 P3 = , （39） P1 + P2 +1 has infinitely many prime solutions. The arithmetic function [6] is æ (P -1)5 +1ö J (w) = ç ÷ ¹ 0. （40） 5 Õ ç P ÷ 3£P£Pi è ø
Since J 5 (w) ® ¥ as w ® ¥, there exist infinitely many primes P1 , P2 , P3 and P4 such that P5 is also a prime. The best asymptotic formula is
p 2 (N,5) = {P1,L, P4 : P1,L,P4 £ N; P5 = prime }
10 J (w)w N 4 = 5 (1+ O(1)). （41） 48f 5 (w) log 5 N
Theorem 14. Diophantine equation
Pn+2 +L+ P2n+1 +1 Pn+1 = , （42） P1 +L+ Pn +1 has infinitely many prime solutions. The arithmetic function [6] is æ (P -1)2n+1 +1ö J (w) = ç ÷ ¹ 0 , （43） 2n+1 Õ ç P ÷ 3£P£Pi è ø
Since J 2n+1 (w) ® ¥ as w ® ¥, there exist infinitely many primes P1 L, P2n such that P2n+1 is also a prime. The best asymptotic formula is
p 2 (N,2n +1) = {P1,L, P2n : P1,L,P2n £ N; P2n+1 = prime }
J (w)w N 2n = 2n+1 (1+ O(1)). （44） 2´(2n)!f 2n+1(w) log 2n+1 N
Theorem 15. Diophantine equation
Pn+2 +L+ P2n+1 Pn+1 = , n ³ 2 , （45） P1 +L+ Pn has infinitely many prime solutions. The arithmetic function [6] is æ (P -1)2n+1 +1 ö J (w) = ç +1÷ ¹ 0, （46） 2n+1 Õ ç P ÷ 3£P£Pi è ø
Since J 2n+1 (w) ® ¥ as w ® ¥, there exist infinitely many primes P1 L, P2n such that P2n+1 is also a prime.
11 The best asymptotic formula [6] is
p 2 (N,2n +1) = {P1,L, P2n : P1,L,P2n £ N; P2n+1 = prime }
J (w)w N 2n = 2n+1 (1+ O(1)). （47） 2´(2n)!f 2n+1(w) log 2n+1 N
Theorem 16. Diophantine equation Pl +L+ P l + P + b m n+2 2n 2n+1 （ ） Pn+1 = l l , 48 P1 +L+ Pn +b has infinitely many prime solutions. The arithmetic function [6] is 2n （ ） J 2n+1(w) = Õ ((P -1) - H(P)) ¹ 0, 49 3£P£Pi Let H (P) denote is the number of solutions of the congruence
l l m l l (q1 +L+ qn + b)qn+1 - qn+2 L- q2n - b º 0 (mod P) , （50） where q j =1,L,P -1, j =1,L,2n.
Since J 2n+1 (w) ® ¥ as w ® ¥, there exist infinitely many primes P1 L, P2n such that P2n+1 is also a prime. The best asymptotic formula [6] is
p 2 (N,2n +1) = {P1,L, P2n : P1,L,P2n £ N; P2n+1 = prime }
J (w)w N 2n = 2n+1 (1+O(1)). （51） (2n)!(m + l)f 2n+1(w) log 2n+1 N
Theorem 17. For every integer m Diophantine equation P +L+ P m = 2n+1 4n , （52） P1 +L+ P2n has infinitely many prime solutions.
12 The arithmetic function [6] is æ (P -1)4n -1 ö J (w) = ç - c(P)÷ ¹ 0 , （53） 4n Õ ç P ÷ 3£P£Pi è ø
(P -1)4n-1 +1 where c(P) = - if P m; c(P) = -1 otherwise. P
Since J 4n (w) ® ¥ as w ® ¥, there exist infinitely many primes P1 L,P4n-1 such that P4n is also a prime. The best asymptotic formula [6] is
p 2 (N,4n) = {P1,L, P4n-1 : P1,L,P4n-1 £ N; P4n = prime }
J (w)w N 4n-1 = 4n (1+O(1)). （54） (4n -1)!f 4n (w) log 4n N
Theorem 18. Diophantine equation
3 3 3 P3 = mP1 + nP2 , (m,n) =1, 2mn, n ¹ ±b , （55） has infinitely many prime solutions. The arithmetic function [1-8] is 2 （ ） J 3 (w) = Õ (P - 3P + 3 - c(P)) ¹ 0 , 56 3£P£Pi
P-1 P-1 where c(P) = 2P -1 if m 3 º n 3 (mod P ); c(P) = -P + 2 if
P-1 P-1 m 3 º/ 2 3 (mod P ); c(P) = -P + 2 if P mn; c(P) = 1 otherwise.
Since J 3 (w) ® ¥ as w ® ¥, there exist infinitely many primes P1 and P2 such that P3 is also a prime. It is the Book proof. The best asymptotic formula [1-8] is
p 2 (N,3) = {P1, P2 : P1,P2 £ N; P3 = prime }
13 J (w)w N 2 = 3 (1+O(1)). （57） 6f 3 (w) log 3 N
3 3 Let m = 1 and n = 2 . From (55) we have P3 = P1 + 2P2 [9].
Theorem 19. Diophantine equation
2 4 P3 = aP1 + cP2 ,(a,c) = 1, 2 ac, a + c ¹ 3d , （58） has infinitely many prime solutions. The arithmetic function [1-8] is 2 （ ） J 3 (w) = Õ (P - 3P + 3 - c(P)) ¹ 0 , 59 3£P£Pi
æ - ac ö æ - ac ö where c(P) = P if ç ÷ =1; c(P) = -P + 2 if ç ÷ = -1 and P ac. è P ø è P ø
Since J 3 (w) ® ¥ as w ® ¥, there exist infinitely many primes P1 and P2 such that P3 is also a prime. It is the Book proof. The best asymptotic formula [1-8] is
p 2 (N,3) = {P1, P2 : P1,P2 £ N; P3 = prime }
J (w)w N 2 = 3 (1+O(1)). （60） 8f 3(w) log 3 N
2 4 Let a = 4 and c =1. From (58) we have P3 = (2P1 ) + P2 [10]. 2 2 2 n 2n Remark. aP1 + bP1P2 + cP2 and aP1 +bP1P2 + cP2 ,n ³ 2 have the same arithmetic function and the same property. If a, b and c have no prime factor in common, they represent infinitely many primes as P1 and P2 run through the positive primes. Gauss proved that there are infinitely many primes of the form x 2 + y2 . It is shown that there are infinitely many primes of the form x 2 + y n ,n ³ 2 [8].
Theorem 20. Two primes represented by P2 - 62 .
14 Suppose that
P2 - 62 = (P + 6)(P -6) . （61）
Form (61) we have two equations
P1 = P + 6 and P2 = P - 6. （62）
The arithmetic function [1-8] is （ ） J 2 (w) = 2 Õ (P -3) ¹ 0, 63 5£P£Pi
Since J 2 (w) ® ¥ as w ® ¥, there exist infinitely many primes P such that
P1 and P2 are primes. It is the Book proof. The best asymptotic formula [1-8] is
p 3 (N,2) = {P : P £ N; P1,P2 = primes }
J (w)w 2 N = 2 (1+O(1)). （64） f 3 (w) log 3 N
Theorem 21. Two primes represented by P3 - 23 . Suppose that
P3 - 23 = (P - 2)(P2 + 2P + 4) . （65）
Form (65) we have two equations
2 P1 = P - 2 and P2 = P + 2P + 4. （66） The arithmetic function [1-8] is
æ æ -3öö J (w) = ç P - 3- ç ÷÷ ¹ 0 , （67） 2 Õ ç P ÷ 5£P£Pi è è øø
Since J 2 (w) ® ¥ as w ® ¥, there exist infinitely many primes P such that
P1 and P2 are primes. The best asymptotic formula [1-8] is
15 p 3 (N,2) = {P : P £ N;P1, P2 = prime }
J (w)w 2 N = 2 (1+ O(1)). （68） 2f 3(w) log 3 N
Theorem 22. Two primes represented by P5 + 25 . Suppose that
P5 + 25 = (P + 2)(P4 - 2P3 + 4P 2 -8P +16) . （69）
Form (69) we have two equations
4 3 2 P1 = P + 2 and P2 = P - 2P + 4P -8P +16 . （70）
The arithmetic function [1-8] is （ ） J 2 (w) = Õ (P -3 - c(P)) ¹ 0 , 71 3£P£Pi where c(P) = 4 if 5(P -1) ; c(P) = 0 otherwise.
Since J 2 (w) ® ¥ as w ® ¥, there exist infinitely many primes P such that
P1 and P2 are primes. The best asymptotic formula [1-8] is
p 3 (N,2) = {P : P £ N;P1, P2 = primes }
J (w)w 2 N = 2 (1+ O(1)). （72） 4f 3(w) log 3 N
Theorem 23. Three primes represented by P4 - 304 .
Suppose that
P4 - 304 = (P + 30)(P - 30)(P 2 +900) . （73）
Form (73) we have three equations
2 P1 = P +30 , P2 = P -30 and P3 = P + 900 . （74）
16 The arithmetic function [1-8] is （ ） J 2 (w) = 8 Õ (P - 3- c(P)) ¹ 0 , 75 7£P£Pi where c(P) = 2 if 4 (P -1) ; c(P) = 0 otherwise.
Since J 2 (w) ® ¥ as w ® ¥, there exist infinitely many primes P such that
P1 , P2 and P3 are primes. The best asymptotic formula [1-8] is
p 4 (N,2) = {P : P £ N;P1,P2 , P3 = primes }
J (w)w 3 N = 2 (1+ O(1)). （76） 2f 4 (w) log 4 N
Theorem 24. Four primes represented by P6 - 426 . Suppose that
P6 - 426 = (P + 42)(P - 42)(P2 + 42P +1764)(P 2 - 42P +1764) .
（77） Form (77) we have four equations
2 2 P1 = P + 42, P2 = P - 42, P3 = P + 42P +1764, P4 = P - 42P +1764 .（78）
The arithmetic function [1-8] is （ ） J 2 (w) = 24 Õ (P - 3- c(P)) ¹ 0, 79 11£P£Pi where c(P) = 4 if 3(P -1) ; c(P) = 0 otherwise.
Since J 2 (w) ® ¥ as w ® ¥, there exist infinitely many primes P such that
P1 , P2 , P3 and P4 are primes. The best asymptotic formula [1-8] is
p 5 (N,2) = {P : P £ N;P1,P2 , P3 ,P4 = primes }
J (w)w 4 N = 2 (1+ O(1)). （80） 4f 5 (w) log 5 N
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REFERENCES [1] C. X. Jiang. On the Yu-Goldbach prime theorem. Guangxi Sciences (in Chinese), 3, (1996) 9-12. [2] C. X. Jiang. Foundations of Santilli’s isonumber theory 1. Algebras, Groups and Geometries 15 (1998) 351-393. MR2000c: 11214. [3] C. X. Jiang. Foundations of Santilli’s isonumber theory 2. Algebras, Groups and Geometries 15 (1998) 509-544. [4] C. X. Jiang. Foundations of Santilli’s isonumber theory. In: Foundamental open problems in sciences at the end of the millennium, T. Gill, K. Liu and E. Trell (Eds) Hadronic Press, USA, (1999) 105-139. [5] C. X. Jiang. Proof of Schinzel’s hypothesis, Algebras Groups and Geometries 18 (2001) 411-420. [6] C. X. Jiang. Foundations of Santilli’s isonumber theory with applications to new cryptograms, Fermat’s theorem and Goldbach’s conjecture. International Academic Press, 2002. www. i-b-r.org [7] C. X. Jiang. Prime theorem in Santilli’s isonumber theory. To appear. [8] C. X. Jiang. Prime theorem in Santilli’s isonumber theory (II). To appear. [9] D. R. Heath-Brown. Primes represented by x 3 + 2y 3 . Acta Math. 186 (2001) 1-84. MR2002b:11122. [10] J. Friedlander and H. Iwaniec. The polynomial x 2 + y4 captures its primes. Ann. of Math. 148 (1998) 945-1040. MR2000c:11150. 2002 Cole prize in number theory and 2002 Ostrowski prize are awarded for this paper.
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