A talk given at Huawei Company (Oct. 31, 2019) and Shenzhen Univ. (Nov. 28, 2019)

Computability and Undecidability

Zhi-Wei Sun

Nanjing University Nanjing 210093, P. R. China [email protected] http://math.nju.edu.cn/∼zwsun

Nov. 28, 2019 Abstract In this talk we introduce the theory of computability and the undecidability results on Hilbert’s Tenth Problem. We will also mention some algorithms concerning primes.

2 / 39 Part I. The Theory of Computability

3 / 39 Primitive recursive functions Let N = {0, 1, 2,...} and call each n ∈ N a natural number. Three Basic Functions: Zero Function: O(x) = 0 (for all x ∈ N). Successor Function: S(x) = x + 1.

Projection Function: Ink (x1,..., xn) = xk (1 6 k 6 n) Composition:

f (x1,..., xn) = g(h1(x1,..., xn),..., hm(x1,..., xn)) Primitive Recursion:

f (x1,..., xn, 0) =g(x1,..., xn),

f (x1,..., xn, y + 1) =h(x1,..., xn, y, f (x1,..., xn, y)) Primitive recursive functions are the basic functions and those obtained from the basic functions by applying composition and primitive recursion a finite number of times. 4 / 39 Acknermann’s function It is easy to see that all primitive recursive functions are computable by our intuition. Skolem’s Claim (1924): All intuitively computable number-theoretic functions are primitive recursive functions. In 1928 Ackermann showed that the following Ackermann function A(m, n) is computable but not primitively recursive.

A(0, n) =n + 1, A(m + 1, 0) =A(m, 1) A(m + 1, n + 1) =A(m, A(m + 1, n)). For example, A(1, 2) =A(0, A(1, 1)) = A(0, A(0, A(1, 0))) =A(0, A(0, A(0, 1))) = A(0, A(0, 2)) =A(0, 3) = 4.

5 / 39 Partial recursive functions

µ-operator:

f (x1,..., xn) = µy(g(x1,..., xn, y) = 0)

means that f (x1,..., xn) is the least natural number y such that g(x1,..., xn, y) = 0. If g(x1,..., xn, y) 6= 0 for all y ∈ N, then f (x1,..., xn) is undefined. Partial recursive functions are the basic functions and those obtained from the basic functions by applying composition and µ-operator a finite number of times.

If a partial recursive function f (x1,..., xn) is defined for all x1,..., xn ∈ N, then f is called a total recursive function. All primitive functions as well as the Ackermann function A(m, n) are total recursive functions.

6 / 39 Church’s Thesis For any partial recursive function f , it is easy to see that if f (x1,..., xn) is defined then the value f (x1,..., xn) is effectively computable. In 1936 A. Turing introduced the notion of Turing machine which is an abstract machine that manipulates symbols on a strip of tape according to a table of rules (i.e., a program). A function f (x1,..., xn) is Turing computable if there is a program according to which the Turing machine with initial inputs x1,..., xn finally stops and yields the value f (x1,..., xn) as outut if f (x1,..., xn) is defined, and never stops if f (x1,..., xn) is undefined. Partial recursive functions and Turing computable functions were proved to be equivalent. Church’s Thesis (1936). If a function f into N with natural number variables is effectively computable by intuition, then it must be a partial recursive function (or a Turing computable function). 7 / 39 Recursively enumerable sets

A subset A of N is said to be an r.e. (recursively enumerable) set (or a semi-decidable set) if the function ( 1 if x ∈ A, f (x) = undefined if x ∈ N \ A.

is a partial recursive function. If A = Dom(f ) for some partial recursive function f , then we may revise the program computing f (x) by letting the output be 1 if f (x) is computed, and thus A is an r.e. set. In view of the above,

A ⊆ N is an r.e. set ⇐⇒ A = Dom(f ) for some partial recursive function f .

8 / 39 Recursively enumerable sets If A is an r.e. set containing an element a, and the program P computes the above function f , then the function ( x if the program P computes f (x) within y steps, g(x, y) = a otherwise

is a partial recursive function with Ran(g) = A.

If A is the range of a partial recursive function h(x1,..., xn), then the function ( 1 if x ∈ Ran(h) = A, f (x) = undefined otherwise,

is a partial recursive function (we may seek for x1,..., xn with h(x1,..., xn) equal to a given x ∈ A), and thus A is an r.e. set. So, a nonempty A ⊆ N is an r.e. set if and only if A = Ran(f ) for some partial recursive function f . 9 / 39 r.e. sets and recursive sets

Enumeration Theorem. There is a partial recursive function ϕ(m, n) such that ϕ0, ϕ1, ϕ2,...

list all the partial recursive functions of one variable. where ϕm is given by ϕm(n) = ϕ(m, n)(n = 0, 1, 2,...).

A set A ⊆ N is called decidable or recursive, if the characteristic function ( 1 if x ∈ A, χA(x) = 0 if x ∈ N \ A. is Turing computable (or recursive).

A set A ⊆ N is recursive if and only if both A and N \ A are r.e. sets.

10 / 39 Halting Problem is undecidable

Theorem. The set K = {x ∈ N : x ∈ Dom(ϕx )} is a nonrecursive r.e. set.

Proof. As the function ϕx (x) = ϕ(x, x) is a partial recursive function, we see that K is an r.e. set. Suppose that K is recursive. Then the function ( ϕ (x) + 1 if x ∈ Dom(ϕ ), f (x) = x x 0 otherwise,

is totally recursive, thus for some m ∈ N we have ϕm = f and hence f (m) = ϕm(m) 6= ϕm(m) + 1 which leads a contradiction.

Let Px be a Turing program computing ϕx . Whether a Turing machine with input x and program Px finally stops, is an undecidable problem which is called the halting problem. 11 / 39 Part II. Solution to Hilbert’s Tenth Problem

12 / 39 Hilbert’s Tenth Problem In 1900, at the Paris conference of ICM, D. Hilbert presented 23 famous mathematical problems. He formulated his tenth problem as follows: Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers. In modern language, Hilbert’s Tenth Problem (HTP) asked for an effective algorithm to test whether an arbitrary polynomial equation

P(z1,..., zn) = 0

(with integer coefficients) has solutions over the ring Z of the integers. However, at that time the exact meaning of algorithm was not known. 13 / 39 Diophantine equations over N and Z Throughout this talk, variables always range over Z. Let P(z1,..., zn) ∈ Z[z1,..., zn]. Then

∃z1 ... ∃zn[P(z1,..., zn) = 0]  Y  ⇐⇒ ∃x1 > 0 ... ∃xn > 0 P(ε1x1, . . . , εnxn) = 0 . ε1,...,εn∈{±1} On the other hand, by Lagrange’s four-square theorem (each m ∈ N can be written as the sum of four squares), we have

∃x1 > 0 ... ∃xn > 0[P(x1,..., xn) = 0] ⇐⇒ ∃u1∃v1∃y1∃z1 ... ∃un∃vn∃yn∃zn 2 2 2 2 2 2 2 2 [P(u1 + v1 + y1 + z1 ,..., un + vn + yn + zn ) = 0] So HTP has the following equivalent form (HTP over N): Is there an algorithm to decide for any polynomial P(x1,..., xn) with integer coefficients whether the Diophantine equation P(x1,..., xn) = 0 has solutions with x1,..., xn ∈ N ? 14 / 39 Diophantine relations and Diophantine sets

A relation R(a1,..., am) with a1,..., am ∈ N is said to be Diophantine if there is a polynomial P(t1,..., tm, x1,..., xn) with integer coefficients such that

R(a1,..., am) ⇐⇒ ∃x1 > 0 ... ∃xn > 0[P(a1,..., am, x1,..., xn) = 0].

A set A ⊆ N is Diophantine if and only if the predicate a ∈ A is Diophantine. It is easy to see that any Diophantine set A is an r.e. set. In fact, for a given element a ∈ A we may search for the natural number solutions of the related Diophantine equation. If it has a solution, then we will find one and let the computer stop and give the output 1. If it has no solution, the computer will never stop.

15 / 39 Davis Daring Hypothesis In 1944 E. L. Post thought that HTP begs for an unsolvability proof, i.e., HTP might be undecidable. In 1949 Martin Davis used G¨odel’scoding idea to obtain that any r.e. set A ⊆ N has the following Davis normal form a ∈ A ⇐⇒ ∃x > 0∀0 6 y 6 x∃z1 > 0 ... ∃zn > 0 [P(a, x, y, z1,..., zn) = 0], where a is a natural number and P is a polynomial with integer coefficients. Davis Daring Hypothesis. Any r.e. set A ⊆ N is Diophantine. Under this hypothesis, for the nonrecursive r.e. set K = {x ∈ N : x ∈ Dom(ϕx )} there is a polynomial P(x, x1,..., xn) such that for any a ∈ N we have a ∈ K ⇐⇒ ∃x1 > 0 ... ∃xn > 0[P(a, x1,..., xn) = 0]. Thus Davis Daring Hypothesis implies that HTP over N is

undecidable. 16 / 39 Systems of Diophantine equations

A system of finitely many Diophantine equations is equivalent to a single Diophantine equation. In fact, if Pi (z1,..., zn) ∈ Z[z1,..., zn] for all i = 1,..., k, then

P1(z1,..., zn) = 0 & ... & Pk (z1,..., zn) = 0 2 2 ⇐⇒ P1 (z1,..., zn) + ... + Pk (z1,..., zn) = 0.

17 / 39 The Davis-Putnam-Robinson Theorem

Theorem (M. Davis, H. Putnam, J. Robinson, Annals of Math. 1961) Any r.e. set is exponential Diophantine. Thus there is no algorithm to decide for any given exponential Diophantine equation whether it has solutions over N.

18 / 39 Julia Robinson’s Hypothesis

n A. Tarski conjectured in 1948 that {2 : n ∈ N} is not a Diophantine set. His PhD student J. Robinson did not succeed in proving this with serious efforts. JR Hypothesis (J. Robinson, 1950). There is a Diophantine relation R(a, b) with a, b ∈ N such that

R(a, b) ⇒ b < aa

and k ∀k > 0∃a > 0∃b > 0[R(a, b)& a < b]. Under this hypothesis, J. Robinson showed that the exponential relation ab = c is Diophantine and hence all r.e. sets are Diophantine. So, the JR Hypothesis implies the negative solution of HTP. J. Robinson tried to prove her JR Hypothesis but got no success. This made her depressed and doubt her Hypothesis. 19 / 39 Matiyasevich’s Theorem

Recall that the Fibonacci sequence (Fn)n>0 defined by

F0 = 0, F1 = 1, and Fn+1 = Fn + Fn−1 (n = 1, 2, 3,...) increases exponentially. In 1970 Yu. Matiyasevich, a 23-year-old Russian, confirmed the JR Hypothesis by showing that the relation y = F2x (with x, y ∈ N) is Diophantine! It follows the exponential relation ab = c (with a, b, c ∈ N, a > 1 and c > 0) is Diophantine, i.e. there exists a polynomial P(a, b, c, x1,..., xn) with integer coefficients such that

b a = c ⇐⇒ ∃x1 > 0 ... ∃xn > 0[P(a, b, c, x1,..., xn) = 0]. This, together with the Davis-Putnam-Robinson work in 1961, led Matiyasevich finally confirm Davis Daring Hypothesis. Matiyasevich’s Theorem (or MDPR Theorem) (1970). Recursively enumerable sets coincide with Diophantine sets. Thus HTP has a negative solution! 20 / 39 Part III. Reduction of Natural Number Unknowns

21 / 39 Small ν with ∃ν over N undecidable n For a set S ⊆ Z we let ∃ over S denote the set of formulas

∃x1 ∈ S ... ∃xn ∈ S[P(x1,..., xn) = 0]

with P(x1,..., xn) ∈ Z[x1,..., xn]. Any nonrecursive r.e. set A has a Diophantine representation:

a ∈ A ⇐⇒ ∃x1 > 0 ... ∃xn > 0[P(x1,..., xn) = 0]. + It is interesting to find the least ν ∈ Z = {1, 2, 3,...} such that ν ∃ over N is undecidable. ν < 200 (Matiyasevich, Summer of 1970) ν 6 35 (J. Robinson, 1970) ν 6 24 (Matiyasevich and Robinson, 1970) ν 6 14 (Matiyasevich and Robinson, 1970) ν 6 13 (Matiyasevich and Robinson, 1973 [Acta Arith. 27(1975)]) ν 6 9 (Matiyasevich, 1975; details in Jones [J. Symbolic Logic, 1982]) 22 / 39 The 9 Unknowns Theorem

The above ideas, together with some other works in the 1975 paper of Matiyasevich and Robinson, led Mtijasevich obtain the following celebrated theorem. 9 Matiyasevich’s 9 Unknowns Theorem: ∃ over N is undecidable! The detailed proof of this theorem appeared in Jones [J. Symbolic Logic, 1982]. ν Up to now, no one has shown that ∃ over N is undecidable for some ν < 9, although A.Baker, Matiyasevich and Robinson all 3 believed that ∃ over N might be undecidable.

23 / 39 ∃ over Z is decidable

Matiyasevich and Robinson [Acta Arith. 27(1975)]: If a0, a1,..., an and z are integers with a0z 6= 0 and Pn n−i i=0 ai z = 0, then

n n n n X n−i X n−1 |z| 6 |a0z | 6 |ai ||z| 6 |ai ||z| i=1 i=1 and hence n X |z| 6 |ai |. i=1 Thus ∃ over N and ∃ over Z are decidable (in polynomial time). 2 It is not known whether ∃ over Z is decidable. But A. Baker proved in 1968 that if P(x, y) ∈ Z[x, y] is homogenous, irreducible and of degree at least three then for any m ∈ Z there is an effective algorithm to determine whether P(x, y) = m for some x, y ∈ Z.

24 / 39 Relative results For any m ∈ Z, by Lagrange’s four-square theorem 2 2 2 2 m > 0 ⇐⇒ ∃z1∃z2∃z3∃z4[m = z1 + z2 + z3 + z4 ]. Thus n 4n ∃ over N is undecidable ⇒ ∃ over Z is undecidable. By the Gauss-Legendre theorem on sums of three squares, 2 2 2 k N \{x + y + z : x, y, z ∈ Z} = {4 (8l + 7) : k, l ∈ N}. 2 2 2 If n ∈ N, then 4n + 1 = (2x) + (2y) + (2z + 1) for some 2 2 2 x, y, z ∈ Z, and hence n = x + y + z + z. Thus, for any m ∈ Z, 2 2 2 m > 0 ⇐⇒ ∃x∃y∃z[m = x + y + z + z]. It follows that n 3n ∃ over N is undecidable ⇒ ∃ over Z is undecidable. 27 Thus ∃ over Z is undecidable by the 9 unknowns theorem, as pointed out by S.P. Tung in [Japan J. Math., 11(1985)]. 25 / 39 A new relation-combining theorem

ν Tung (1985) asked whether ∃ over Z is undecidable for some ν < 27. New Relation-Combining Theorem (Z.-W. Sun [Z. Math. Logik Grundlag. Math. 38(1992)]): Let A1,..., Ak , B, C1,..., Cn, D, E be integers with D 6= 0. Then

A1,..., Ak ∈  ∧ B 6= 0 ∧ C1,..., Cn > 0 ∧ D | E ⇐⇒ ∃z1 ... ∃zn+2[P(A1,..., Ak , B, C1,..., Cn, D, E, z1,..., zn+2) = 0],

where P is a suitable polynomial with integer coefficients. This implies that

n 2n+2 ∃ over N is undecidable ⇒ ∃ over Z is undecidable.

20 So ∃ over Z is undecidable by the 9 unknowns theorem.

26 / 39 ∃11 over Z is undecidable 11 In 1992, I announced that ∃ over Z is undecidable. To achieve this goal, unlike others I did not simply use the relative result, instead I adapt the deep proof of the 9 unknowns theorem and made suitable variants so that we can use integer variables instead of natural number variables. My starting point is the use of Lucas sequences with integer indices instead of the usual natural number indices. I published this initial step in Sci. China Ser. A 35(1992). 11 The whole proof of the undecidability of ∃ over Z is very sophisticated. It appeared in my PhD thesis in 1992. During 1992-2016, despite that many mathematicians wanted to see my detailed proof, I did not write an English version of that, since I was frequently busy with my new discoveries. After 25 years have passed, I finally spent time to write an English 11 paper which contains the undecidability of ∃ over Z as well as my new discoveries related to HTP. The preprint is now publicly available from http://arxiv.org/abs/1704.03504 27 / 39 Part III. Some Algorithms involving Primes

28 / 39 Initial Results on Discriminants Theorem 1 (L. K. Arnold, S. J. Benkoski and B. J. McCabe [Amer. Math. Monthly 92 (1985), no. 4]). Let n > 4 be an integer. Then the least positive integer m (denoted by D(n)) such that 11, 22,..., n2 are distinct modulo m, is min{m > 2n : m = p or m = 2p with p an odd prime}.

Remark. The range of D(n) does not contain those primes p = 2q + 1 with q an odd prime. The theorem arose from consideration of a problem in computer simulation. The problem may be described as developing a method to determine quickly the square roots of a long sequence of integers. For a long sequence 12, 22,..., n2 of squares, you may √ 2 use the function A(x) = x (1 6 x 6 n ) to get the square roots. This needs an array of size 1 × n2. If 12, 22,..., n2 are distinct √ modulo m, then letting A(r) = x where r ∈ {1,..., m} and r ≡ x (mod m). The modulo procedure is , in general, much faster than the square root procedure. 29 / 39 Generate all primes in a combinatorial manner

+ Theorem 1 (Sun, Feb. 29, 2012) (i) For n ∈ Z let S(n) denote the smallest integer m > 1 such that those 2k(k − 1) mod m for k = 1,..., n are pairwise distinct. Then S(n) is the least prime greater than 2n − 2. + (ii) For n ∈ Z let T (n) denote the least integer m > 1 such that those k(k − 1) mod m with 1 6 k 6 n are pairwise distinct. Then we have

T (n) = min{m > 2n − 1 : m is a prime or a positive power of 2}.

Remark. (a) The range of S(n) is exactly the set of all primes! (b) I proved that the least positive integer m such that those k
2 = k(k − 1)/2 (k = 1,..., n) are pairwise distinct modulo m, is just the least power of two not smaller than n.

30 / 39 Another theorem

+ Theorem 2 (Sun, March 2012) (i) Let d ∈ {2, 3} and n ∈ Z . Then the smallest positive integer m such that those k(dk − 1) (k = 1,..., n) are pairwise distinct modulo m, is the least power of d not smaller than n. (ii) Let n ∈ {4, 5,...}. Then the least positive integer m such that 18k(3k − 1) (k = 1,..., n) are pairwise distinct modulo m, is just the least prime p > 3n with p ≡ 1 (mod 3). Remark. We are also able to prove some other similar results; for + example, for each n > 5 the least m ∈ Z such that those 18k(3k + 1) (k = 1,..., n) are pairwise distinct modulo m, is just the first prime p ≡ −1 (mod 3) after 3n.

31 / 39 Alternating sums of primes

Let pn be the nth prime and define

n−1 sn = pn − pn−1 + ··· + (−1) p1.

Note that

n n X X s2n = (p2k − p2k−1) > 0, s2n+1 = (p2k+1 − p2k ) + p1 > 0. k=1 k=1

Here are values of s1,..., s16:

2, 1, 4, 3, 8, 5, 12, 7, 16, 13, 18, 19, 22, 21, 26, 27.

The sequence 0, s1, s2,... were first introduced by N.J.A. Sloane and J.H. Conway (see A008347 at OEIS).

It is not difficult to show that those sn (n = 1, 2, 3,...) are pairwise distinct.

32 / 39 An amazing recurrence for primes

The following surprising conjecture on recurrence for primes allows us to compute pn+1 in terms of p1,..., pn. Conjecture (Sun, March 28, 2012). For any positive integer n 6= 1, 2, 4, 9, the (n + 1)th prime pn+1 is the least positive integer m such that 2 2 2s1 ,..., 2sn are pairwise distinct modulo m. 5 Remark. I have verified the conjecture for n 6 2 × 10 , and proved 2 2 that 2s1 ,..., 2sn are indeed pairwise distinct modulo pn+1. A Related Conjecture (Sun, March 26, 2012). The least integer 2 m > 1 such that 2Sk (k = 1,..., n) are pairwise distinct modulo 2 Pk m is a prime smaller than n unless n | 6, where Sk = j=1 pj .

33 / 39 Conjecture on alternating sums of consecutive primes Conjecture (Sun, April 2-3, 2012). For any positive integer m, there are consecutive primes p ,..., p (k n) not exceeding √ k n 6 2m + 2.2 m such that

n−k m = pn − pn−1 + ··· + (−1) pk . Examples.

10 = 17 − 13 + 11 − 7 + 5 − 3; 20 = 41 − 39 + 37 − 31 + 29 − 23 + 19 − 17 + 13 − 11;

2382 = p652 − p651 + ··· + p44 − p43, √ p652 = 4871 = b2 · 2382 + 2.2 2382c. The conjecture has been verified for m up to 109. Most known results on primes are about local properties of primes, not about relations of primes. Prize. I would like to offer 1000 US dollars for the first proof. 34 / 39 How to solve x 2 ≡ a (mod p)? Let p be an odd prime and a be any quadratic residue modulo p. How to solve the congruence x2 ≡ a (mod p) quickly? Tonelli-Shanks Algorithm. Knowing a quadratic nonresidue 2 d ∈ Z mod p, one can solve x ≡ a (mod p) in polynomial time: s + Write p − 1 = 2 t with s, t ∈ Z and 2 - t, and find even integers m 2s−i t m1,..., ms with (ad i ) ≡ 1 (mod p) for all i = 1,..., s in the following way: m1 := 0, and after those m1,..., mi (with i 1 6 i < s) have been chosen we select mi+1 ∈ {mi , mi + 2 } such s−i−1 s−i−1 that (admi+1 )2 t ≡ 1 (mod p). Note that ((admi )2 t )2 ≡ 1 s−i−1 (mod p) and hence (admi )2 t ≡ ±1 (mod p). If s−i−1 (admi )2 t ≡ −1 (mod p), then i s−1−i s−1 (admi +2 )2 t ≡ −d2 t = −d(p−1)/2 ≡ 1 (mod p). As (adms )t ≡ 1 (mod p), we have x2 ≡ a (mod p) with x = ±a(t+1)/2(dt )ms /2. Remark. In 1985 R. Schoof gave a polynomial (depending on a) 2 algorithm to solve the congruence x ≡ a (mod p) with a given. 35 / 39 Find a quadratic nonresidue modulo a prime

However, there is no known deterministic, polynomial time algorithm for finding a quadratic nonresidue d modulo the odd prime p. Under the Extended Riemann Hypothesis for algebraic fields, it can be shown that there is a positive quadratic nonresidue d < 2 log2 p; and so an exhaustive search to this limit succeeds in finding a quadratic nonresidue in polynomial time. As Fibonacci numbers grow exponentially, our following conjecture is particularly interesting since it implies that we can find square roots for quadratic residues modulo any odd prime p in deterministic, polynomial time. Conjecture (Z.-W. Sun, 2014) For any integer n > 4, there is a Fibonacci number f < n/2 with x2 ≡ f (mod n) for no integer x. Remark. This can be reduced to the case with n = p prime. We have verified that for any prime 3 < p < 3 × 109 there is a Fk Fibonacci number Fk < p/2 with ( p ) = −1.

36 / 39 The partition function p(n) A partition of a positive integer n is a way to write n as a sum of positive integers with the order of addends ignored. The partition function p(n) denotes the total number of partitions of n. Example. p(5) = 7 since 5=1+4=2+3=1+1+3=1+2+2 =1 + 1 + 1 + 2 = 1 + 1 + 1 + 1 + 1. Hardy-Ramanujan Formula: √ eπ 2n/3 p(n) ∼ √ as n → +∞, 4 3n and hence √ r2 log p(n) ∼ c n with c = π . 3 Note that p(n) grows eventually faster than any polynomial in n but slightly slower than 2n. Mathematica could compute p(n) quickly. 37 / 39 On primes related to p(n) Conjecture 1 (Sun, 2014-02-27). Let n be any positive integer. Then one of the n numbers p(n) + 1, p(n) + 2,..., p(n) + n is prime. Conjecture 2 (Sun, 2014-02-28). Let n > 1 be an integer. Then p(n) + p(k) − 1 is prime for some 0 < k < n. Conjecture 3 (Sun, 2014-03-13). Let n > 3 be an integer. Then p(n + k) + 1 is prime for some k = 1,..., n. Conjecture 4 (Sun, 2014-03-12). Let n > 1 be an integer. Then there exists a number k ∈ {1,..., n − 1} such that kp(n)(p(n) − 1) + 1 is prime. Also, we may replace kp(n)(p(n) − 1) + 1 by p(k)p(n)(p(n) − 1) + 1 or p(k)p(n)(p(n) + 1) − 1. These conjectures might be helpful in finding large primes for the use of RSA. 38 / 39 References For main sources of my work mentioned here, you may look at: 1. Z.-W. Sun, Reduction of unknowns in Diophantine representations, Sci. China Math. 35(1992), 257–269. 2. Z.-W. Sun, A new relation-combining theorem and its application, Z. Math. Logik Grundlag. Math. 38(1992), 209-212. 3. Z.-W. Sun, Further results on Hilbert’s tenth problem, arXiv:1704.03504, http://arxiv.org/abs/1704.03504. 4. Z.-W. Sun, On functions taking only prime values, J. Number Theory 133 (2013), 2794–2812. 5. Z.-W. Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28–Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Thank you! 39 / 39