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```CHAP. XVIII]                             NUMBBB OF PBIMES.                                          429
NUMBEK OF PEIMES BETWEEN ASSIGNED LIMITS.
Formula (5) of Legendre in Ch. V implies that if 0, X,... are the primes ^ Vn, the number of primes £ n and > Vn is one less (if unity be counted a prime) than
Statements or proofs of this result have been given by C. J. Hargreave,205 E. de Jonquieres,206 R. Lipschitz,207 J. J. Sylvester,208 E. Catalan,209 F. Ro-gel,210 J. Hammond211 with a modification, H. W. Curjel,2110 S. Johnsen,212 and L. Kronecker.218
E. Meissel214 proved that if B(m) is the number of primes (including unity) ^m and if
E. Meissel216 wrote <3>(w, n) for Legendre's formula for the number of integers ^ m which are divisible by no one of the first n primes pi = 2, . . . , pn.
Then                                                     /Tw~l         \
<i>(m, n)=\$(w, n— 1)— \$(   — Ln— !]•
\ Lpn J          /
Let 0(w) be the number of primes gra. Setn+M=^(Vw), n=0(-^5n).  Then
-l- S
which is used to compute B(m) for w = fc-106, A; = 1/2, 1, 10. Meissel216 applied his last formula to find 0(108). Lionnet2160 stated that the number of primes between A and 2 A is
N. V. Bougaief217 obtained from 0(n)+0(n/2)+0(n/3) + . . . =2[n/p], by inversion (Ch. XIX),
where a, 6, ... range over all primes.
2°'Lond. Ed. Dub. Phil. Mag., (4), 8, 1854, 118-122.
J08Compte8 Rendus Paris, 95, 1882, 1144, 1343; 96, 1883, 231.
™IUd., 95, 1882, 1344-6; 96, 1883, 58-61, 114-5, 327-9.
208/btrf., 96, 1883, 463-5; Coll. Math. Papers, 4, p. 88.
208M6m. Soc. Roy. Sc. de Ltege, (2), 12, 1885, 119; Melanges Math., 1868, 133-5.
J1"Archiv Math. Phys., (2), 7, 1889, 381-8.                        '"Messenger Math., 20, 1890-1, 182.
""'Math. Quest. Educ. Times, 67, 1897, 27.
212Nyt Tidsskrift for Mat., Kjobenhavn, 15 A, 1904, 41-4.
21'Vorlesungen tiber Zahlentheorie, I, 1901, 301-4.               "Mour. ftir Math., 48, 1854, 310-4.
215Math. Ann., 2, 1870, 636-642.   Outline in Mathews' Theory of Numbers, 273-8, and in G.
Wertheim'e Elemente der Zahlentheorie, 1887, 20-25. 21fl/6id., 3, 1871, 523-5.   Corrections, 21, 1883, 304. 21«aNouv. Ann. Math., 1872, 190.    Cf. Landau, (4), 1, 1901, 281-2. 217Bull. ec. math, aetr., 10, 1, 1876, 16.    Mat. Sbornik (Math. Soc. Moscow), 6, 1872-3, I, 180.```