MathSciNet infinitude of prime

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1 0 π(x) x π(x) > c log log x π(x) > c log x c MathSciNet infinitude of prime 2018/11/ arxiv Romeo Meštrović, Euclid s theorem on the infinitude of primes: a historical survey of its proofs (300 B.C. 2012) = p 1 < p 2 < < p r P = p 1 p 2 p r + 1 p P p 1, p 2,..., p r P = p 1 p 2 p r + 1 p p 1, p 2,..., p r R 2 R < p R R + 1 1

2 2 2 2 = p 1 < p 2 < < p r R P = p 1 p 2 p r + 1 p P p 1, p 2,..., p r P = p 1 p 2 p r + 1 p p R r + 1 R R + 1 p 1, p 2,..., p r R p > R R R + 1 R R/ R R/2 + 1 p 1 p 2 p r = p 1 p 2 p r + 1 p 1 p 2 p r + 1 P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, New York, 1996, p. 3. L. E. Dickson, History of the Theory of Numbers, Vol. I: Divisibility and Primality, Carnegie Institution of Washington, 1919 (reprint: Dover Publication, New York, 2005), p = p 1 < p 2 < < p r N = p 1 p 2 p r > 2 2 N N 1 > p r N 1 p i p i N (N 1) = 1 p 1 p 2 p r + 1 p 1 p 2 p r 1 R 2 R < p R R/2+1 1 p

3 3 3 P. Ribenboim,, p. 4 L. E. Dickson,, p p 1, p 2,..., p r N = p 1 p 2 p r N = mn(m, n 1) N p i m n p i m + n m + n m + n > 1 P. Ribenboim,, p. 4 L. E. Dickson,, p F n = 2 2n + 1(n 0) 2 F m 2 = 2 2n 1 = (2 2n 1 + 1)(2 2n 1 1) = F 0 F 1 F m 1 F m F n (m > n) 2 = F m (F m 2) F n q 1 F 1 q 2 F 2 q 1, q 2, F 0 = 3, F 1 = 5, F 2 = 17, F 3 = 257, F 4 = F 5 = = 641 P. Ribenboim,, p.p. 4 5 L. E. Dickson,, p. 413

4 5 4 5 m n = m + 1 n (n!)i + 1(i = 1, 2,..., n) 2 1 i < j n j = i+d ((n!)i+1, (n!)j+1) = ((n!)i+1, (n!)d) = 1 i = 1, 2,..., n p i (n!)i + 1 p 1, p 2,..., p n n n (n!)n + 1 n P. Ribenboim,, p.p. 5 6 p 1, p 2,..., p n 1 k=0 1/pk i 1/(1 1/p i ) k=0 1/pk i = k i 1/( n i=1 pk i n i=1 1/(1 1/p i) = n i=1 i ) p 1, p 2,..., p n n i=1 pk i i 1 n=1 1/n n i=1 1/(1 1/p i) n=1 1/n n=1 1/n 1 + 1/2 + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8) + n=1 1/ns = n i=1 1/(1 1/ps i ) ζ ζ(s) = n=1 1/ns N N N n=1 1/n p N 1/(1 1/p)

5 7 5 log p N 1/(1 1/p) = p N log(1 1/p) log(1 1/p) = m=1 1/(mpm ) m=1 1/(pm ) = 1/p+1/p(p 1) < 1/p+1/(p 1) 2 log N n=1 1/n p N 1/p + 1/(p 1)2 p N 1/p + n 1/n2 p N 1/p log N n=1 1/n c log log N C c, C P. Ribenboim,, p.p. 6 7 L. E. Dickson,, p p 1 = 2, p 2 = 3,..., p j x n = n 2 1m m m = 2 b 1 3 b2 p b j j b i j n 2 1 n x m n 1 x 1/2 x 2 j x 1/2 j (log x)/(2 log 2) x (log x)/(2 log 2) p>m 1/p < 1/2 M N p>m N/p < N/2 N 2 N 1 N M N 2 N N 1 p>m N/p < N/2 N 2 < 2 M N 1/2 N N = N 1 + N 2 < N/2 + 2 M N 1/2 < N M N = 2 2M+2 N 1 M<p N N/p N 1 = N N 2 N 2 M N 1/2 = 2 2M+1 = N/2 M<p 2 2M+2 1/p > 1/2 G. H. Hardy and E. M. Wright, D. R. Heath-Brown, J. Silverman, A. Wiles, An Introduction to the Theory of Numbers, the 6th edition, Oxford University Press, 2008, Section 2.6.

6 8 6 8 x > 1 x < p < 2x a(p, N) p N! p N a(p, N) = N/p + N/p 2 + < N/(p 1) p N a(p, N) = 0 p N (log p)/(p 1) > p a(p, N)(log p)/n = (1/N) log(n!) > (log N) 1 p N (log p)/(p 1) N G. H. Hardy and E. M. Wright, D. R. Heath-Brown, J. Silverman, A. Wiles, An Introduction to the Theory of Numbers, the 6th edition, Oxford University Press, 2008, Section As mentioned above, any infinite sequence of pairwise coprime integers shows the infinite of primes. x i (i = 0, 1,...) x i x i+1 gcd(x i, x i+1 /x i ) = 1 a i = x i+1 /x i (i = 0, 1,...) 2 f(x) = x 2 + x + 1 f(n 2 ) = n 4 + n = (n 2 + n + 1)(n 2 n + 1) = f(n)f( n) n 2 + n + 1 gcd(n 2 + n + 1, n 2 n + 1) = gcd(n 2 + n + 1, 2n) = 1 x i = f(2 2m ) x m (m = 0, 1,...) x m = 2 pm 1(m = 0, 1,...) q x m x m+1 /x m = ((x m +1) p 1)/((x m +1) 1) = 1+(x m +1)+ +(x m +1) p 1 p

7 10 7 (mod q) q x m, x m+1 /x m q = p x m = (2 m ) p 1 1 (mod p) p x m gcd(x m, x m+1 /x m ) = 1 2 pm 1 p m 1 p m 1 k k 1 k 1 S. Srinivasan, On infinitude of primes, Hardy Ramanujan J. 7 (1984), n, k (1+n) k < 2 n p 1 = 2, p 2 = 3,..., p r 2 n 1 m 2 n m = 2 e 1 3 e2 p er r m 2 n e i n e 1, e 2,..., e r (n + 1) r (1 + n) k < 2 n (1 + n) r r > k k 1 + 2k 2 < 2 2k (1 + 2k 2 ) k < 2 2k2 k n = 2k 2 n, k 2 n = 4 k2 k + 1 P. Ribenboim,, p. 7 L. E. Dickson,, p n=1 (1/n2 ) 2 π 2 /6 n=1 1 n = n=3 1 n < n=3 1 n(n 1) = n=3 n=1 (1/n2 ) < 7/4 ( 1 n 1 1 ) = 5 n = 7 4

8 12 8 δ = 2 n=1 (1/n2 ) δ > 1/4 X p 1, p 2,..., p r X X m 2 r X m d 2 X/d 2 X m d=2 (X/d2 ) = X( d=1 (1/d2 ) 1) = X(1 δ) X 2 r + X(1 δ) 2 r δx N/4 r > (log X/ log 2) 2 X d 2 X P. Ribenboim,, p. 8 L. E. Dickson,, p p 1 < p 2 < < p r t N = p t r N m r (f 1, f 2,..., f r ) m = p f 1 1 p f 2 2 p fr r E = (log p r )/(log p 1 ) p f i 1 p f i i m N = p t r i f i te N 1 f i te r (f 1, f 2,..., f r ) p t r = N (te + 1) r t r (E + 1) r t P. Ribenboim,, p. 9 L. E. Dickson,, p Boije af Gennäs X 2, 3,..., p n X e i (i = 1, 2,..., n) P = 2 e 1 3 e2 p en n δ P/δ Q = P/δ δ > 1 δ Q X Q X X

9 14 9 L. E. Dickson,, p p 1 = 2, p 2 = 3,..., p t P = t i=1 p i, Q = t i=2 p i x = (P + P 2 + 4)/2 = Q + Q x = [p, p,...] Q p 1 = 2 2 Q Q = 2 2l+1 Q 2 2(2 l ) 2 = 1 Q/2 l 2 [1, 2, 2,...] [1, 2, 2,...] m B m B 0 = 1, B 1 = 2, B m+1 = 2B m + B m 1 m B m Q l = 0 Q = 1 Q > 2 Q Q C. W. Barnes, The infinitude of primes; a proof using continued fractions, L Enseignement Math. (2) 22 (1976), t p 1, p 2,..., p t m/n = t i=1 1/p i 1/2 + 1/3 + 1/5 > 1 m/n > 1 m > n 1 m p i p i p 1 p 2 p i 1 p i+1 p t L. E. Dickson,, p. 414

10 A 0, A 1, A 2 2 n 3 A n = A 0 A 1 A n 3 A n 1 + A n 2 A 0, A 1,..., A n 2 p gcd(a n, A n 2 ) p A n A n 2 = A 0 A 1 A n 3 A n 1 i 1, i 2 p gcd(a n i, A n 2 ) p gcd(a n, A n j ) for some j 1, j 2 p A n A 0 A 1 A n 3 A n 1 = A n 2 p gcd(a n j, A n 2 ) A 0, A 1,..., A n 2 n 3 A n > 1 A 0, A 1, A 0 = b 0, A 1 = b 0 b 1 +1, A 2 = b 0 b 1 b 2 +b 0 +b 2 and b n = A 0 A 1 A n 3 b i (i = 0, 1,...) A n [b 0, b 1, b 2,...] V. C. Harris, Another proof of the infinitude of primes, Amer. Math. Monthly 63 (1956), k p 1, p 2,..., p k N k (e 1, e 2,..., e k ) p e 1 1 p e 2 2 p e k k N e 1 log p 1 + e 2 log p e k log p k log N N N (log N) k /(k! log p 1 log p 2 log p k ) N c(log N) k c N N p 1, p 2,..., p k N c = 1/(k! log 2) N (log N) k /(k! log 2) k > c log N c N c log N Paul R. Chernoff, A Lattice Point Proof of the Infinitude of Primes. Math. Mag. 38 (1965), 208.