Abstract
It is proved that $$ \liminf_{n\to\infty}(p_{n+1}p_n)<7\times 10^7, $$ where $p_n$ is the $n$th prime.
Our method is a refinement of the recent work of Goldston, Pintz and Yıldırım on the small gaps between consecutive primes. A major ingredient of the proof is a stronger version of the BombieriVinogradov theorem that is applicable when the moduli are free from large prime divisors only, but it is adequate for our purpose.

[1] E. Bombieri, J. B. Friedlander, and H. Iwaniec, "Primes in arithmetic progressions to large moduli," Acta Math., vol. 156, iss. 34, pp. 203251, 1986.
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[2] E. Bombieri, J. B. Friedlander, and H. Iwaniec, "Primes in arithmetic progressions to large moduli. II," Math. Ann., vol. 277, iss. 3, pp. 361393, 1987.
@article {2, MRKEY = {0891581},
AUTHOR = {Bombieri, E. and Friedlander, J. B. and Iwaniec, H.},
TITLE = {Primes in arithmetic progressions to large moduli. {II}},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {277},
YEAR = {1987},
NUMBER = {3},
PAGES = {361393},
ISSN = {00255831},
CODEN = {MAANA},
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MRNUMBER = {0891581},
MRREVIEWER = {D. R. HeathBrown},
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[3] E. Bombieri, J. B. Friedlander, and H. Iwaniec, "Primes in arithmetic progressions to large moduli. III," J. Amer. Math. Soc., vol. 2, iss. 2, pp. 215224, 1989.
@article {3, MRKEY = {0976723},
AUTHOR = {Bombieri, E. and Friedlander, J. B. and Iwaniec, H.},
TITLE = {Primes in arithmetic progressions to large moduli. {III}},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {2},
YEAR = {1989},
NUMBER = {2},
PAGES = {215224},
ISSN = {08940347},
MRCLASS = {11N13},
MRNUMBER = {0976723},
MRREVIEWER = {D. R. HeathBrown},
DOI = {10.2307/1990976},
ZBLNUMBER = {0674.10036},
} 
[4] P. Deligne, "La conjecture de Weil. I," Inst. Hautes Études Sci. Publ. Math., vol. 43, pp. 273307, 1974.
@article {4, MRKEY = {0340258},
AUTHOR = {Deligne, Pierre},
TITLE = {La conjecture de {W}eil. {I}},
JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
VOLUME = {43},
YEAR = {1974},
PAGES = {273307},
ISSN = {00738301},
MRCLASS = {14G13},
MRNUMBER = {0340258},
MRREVIEWER = {Nicholas M. Katz},
ZBLNUMBER = {0287.14001},
DOI = {10.1007/BF02684373},
} 
[5] E. Fouvry and H. Iwaniec, "Primes in arithmetic progressions," Acta Arith., vol. 42, iss. 2, pp. 197218, 1983.
@article {5, MRKEY = {0719249},
AUTHOR = {Fouvry, E. and Iwaniec, H.},
TITLE = {Primes in arithmetic progressions},
JOURNAL = {Acta Arith.},
FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
VOLUME = {42},
YEAR = {1983},
NUMBER = {2},
PAGES = {197218},
ISSN = {00651036},
CODEN = {AARIA9},
MRCLASS = {10H15 (10H20)},
MRNUMBER = {0719249},
MRREVIEWER = {Matti Jutila},
ZBLNUMBER = {0517.10045},
URL = {http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournalarticleaav42i2p197bwm?q=bwmeta1.element.bwnjournalnumberaa19821983422;5&qt=CHILDRENSTATELESS},
} 
[6] J. B. Friedlander and H. Iwaniec, "Incomplete Kloosterman sums and a divisor problem," Ann. of Math., vol. 121, iss. 2, pp. 319350, 1985.
@article {6, MRKEY = {0786351},
AUTHOR = {Friedlander, John B. and Iwaniec, Henryk},
TITLE = {Incomplete {K}loosterman sums and a divisor problem},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {121},
YEAR = {1985},
NUMBER = {2},
PAGES = {319350},
ISSN = {0003486X},
CODEN = {ANMAAH},
MRCLASS = {11N25 (11L40 11N69)},
MRNUMBER = {0786351},
MRREVIEWER = {D. R. HeathBrown},
DOI = {10.2307/1971175},
ZBLNUMBER = {0572.10029},
} 
[7] D. A. Goldston, J. Pintz, and C. Y. Yildirim, "Primes in tuples. I," Ann. of Math., vol. 170, iss. 2, pp. 819862, 2009.
@article {7, MRKEY = {2552109},
AUTHOR = {Goldston, Daniel A. and Pintz, J{á}nos and Yıldırım, Cem Y.},
TITLE = {Primes in tuples. {I}},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {170},
YEAR = {2009},
NUMBER = {2},
PAGES = {819862},
ISSN = {0003486X},
CODEN = {ANMAAH},
MRCLASS = {11N05 (11N36)},
MRNUMBER = {2552109},
MRREVIEWER = {S. W. Graham},
DOI = {10.4007/annals.2009.170.819},
ZBLNUMBER = {1207.11096},
} 
[8] D. A. Goldston, J. Pintz, and C. Y. Yildirim, "Primes in tuples. II," Acta Math., vol. 204, iss. 1, pp. 147, 2010.
@article {8, MRKEY = {2600432},
AUTHOR = {Goldston, Daniel A. and Pintz, J{á}nos and Yıldırım, Cem Yal{ç}in},
TITLE = {Primes in tuples. {II}},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {204},
YEAR = {2010},
NUMBER = {1},
PAGES = {147},
ISSN = {00015962},
CODEN = {ACMAA8},
MRCLASS = {11N05 (11N36)},
MRNUMBER = {2600432},
MRREVIEWER = {S. W. Graham},
DOI = {10.1007/s1151101000449},
ZBLNUMBER = {1207.11097},
} 
[9] D. R. HeathBrown, "Prime numbers in short intervals and a generalized Vaughan identity," Canad. J. Math., vol. 34, iss. 6, pp. 13651377, 1982.
@article {9, MRKEY = {0678676},
AUTHOR = {HeathBrown, D. R.},
TITLE = {Prime numbers in short intervals and a generalized {V}aughan identity},
JOURNAL = {Canad. J. Math.},
FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de Mathématiques},
VOLUME = {34},
YEAR = {1982},
NUMBER = {6},
PAGES = {13651377},
ISSN = {0008414X},
CODEN = {CJMAAB},
MRCLASS = {10H15 (10H25)},
MRNUMBER = {0678676},
MRREVIEWER = {Kenneth A. Jukes},
DOI = {10.4153/CJM19820959},
ZBLNUMBER = {0478.10024},
} 
[10] D. R. HeathBrown, "The divisor function $d_3(n)$ in arithmetic progressions," Acta Arith., vol. 47, iss. 1, pp. 2956, 1986.
@article {10, MRKEY = {0866901},
AUTHOR = {HeathBrown, D. R.},
TITLE = {The divisor function {$d\sb 3(n)$} in arithmetic progressions},
JOURNAL = {Acta Arith.},
FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
VOLUME = {47},
YEAR = {1986},
NUMBER = {1},
PAGES = {2956},
ISSN = {00651036},
CODEN = {AARIA9},
MRCLASS = {11N25 (11N37)},
MRNUMBER = {0866901},
MRREVIEWER = {A. J. Hildebrand},
ZBLNUMBER = {0549.10034},
URL = {http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournalarticleaav47i1p29bwm?q=bwmeta1.element.bwnjournalnumberaa1986471;2&qt=CHILDRENSTATELESS},
} 
[11] H. Iwaniec, "A new form of the error term in the linear sieve," Acta Arith., vol. 37, pp. 307320, 1980.
@article {11, MRKEY = {0598883},
AUTHOR = {Iwaniec, Henryk},
TITLE = {A new form of the error term in the linear sieve},
JOURNAL = {Acta Arith.},
FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
VOLUME = {37},
YEAR = {1980},
PAGES = {307320},
ISSN = {00651036},
CODEN = {AARIA9},
MRCLASS = {10H30},
MRNUMBER = {0598883},
MRREVIEWER = {H.E. Richert},
ZBLNUMBER = {0444.10038},
URL = {http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournalarticleaav37i1p307bwm?q=bwmeta1.element.bwnjournalnumberaa1980371;26&qt=CHILDRENSTATELESS},
} 
[12] P. Shiu, "A BrunTitchmarsh theorem for multiplicative functions," J. Reine Angew. Math., vol. 313, pp. 161170, 1980.
@article {12, MRKEY = {0552470},
AUTHOR = {Shiu, P.},
TITLE = {A {B}run{T}itchmarsh theorem for multiplicative functions},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
VOLUME = {313},
YEAR = {1980},
PAGES = {161170},
ISSN = {00754102},
CODEN = {JRMAA8},
MRCLASS = {10H25},
MRNUMBER = {0552470},
MRREVIEWER = {A. I. Vinogradov},
DOI = {10.1515/crll.1980.313.161},
ZBLNUMBER = {0412.10030},
} 
[13] K. Soundararajan, "Small gaps between prime numbers: the work of GoldstonPintzY\i ld\i r\i m," Bull. Amer. Math. Soc., vol. 44, iss. 1, pp. 118, 2007.
@article {13, MRKEY = {2265008},
AUTHOR = {Soundararajan, K.},
TITLE = {Small gaps between prime numbers: the work of {G}oldston{P}intz{Y}\i ld\i r\i m},
JOURNAL = {Bull. Amer. Math. Soc.},
FJOURNAL = {American Mathematical Society. Bulletin. New Series},
VOLUME = {44},
YEAR = {2007},
NUMBER = {1},
PAGES = {118},
ISSN = {02730979},
CODEN = {BAMOAD},
MRCLASS = {11N05 (1102 11N36)},
MRNUMBER = {2265008},
MRREVIEWER = {D. R. HeathBrown},
DOI = {10.1090/S0273097906011426},
ZBLNUMBER = {1193.11086},
}