Limit of Difference between Consecutive Prime Numbers/Historical Note
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Historical Note on Limit of Difference between Consecutive Prime Numbers
Gregorio Ricci-Curbastro showed that $E \le \dfrac {15} {16}$.
Enrico Bombieri and Harold Davenport showed in $1966$ that $E \le \dfrac {2 + \sqrt 3} 8 \approx 0 \cdotp 46650 \ldots$
In their $1983$ work Les Nombres Remarquables, François Le Lionnais and Jean Brette report that Pil'Tai deduced in $1972$ that $E \le \dfrac {2 \sqrt 2 - 1} 4 \approx 0 \cdotp 45706 \ldots$
However, it is not clear who Pil'Tai is or was, and this has not been corroborated.
In $1973$ Martin Neil Huxley showed that $E \le \dfrac 1 4 + \dfrac \pi {16} \approx 0 \cdotp 44634 \, 95408 \ldots$
Sources
- 1973: M.N. Huxley: Small differences between consecutive primes (Mathematica Vol. 20: p. 24)
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,44634 95408 \ldots$