Squuse number

The Skewes number is the smallest positive integer. $n$ ${\ displaystyle n}$ $n$ , such that, starting with it, the inequality $\pi (n)<\operatorname {Li} (n)$ ${\ displaystyle \ pi (n) <\ operatorname {Li} (n)}$ ${\ displaystyle \ pi (n) <\ operatorname {Li} (n)}$ stops running where $\pi (n)$ ${\ displaystyle \ pi (n)}$ $\ pi (n)$ - distribution function of primes , $\operatorname {Li} (n)=\int \limits _{2}^{n}{\frac {dt}{\ln(t)}}$ ${\ displaystyle \ operatorname {Li} (n) = \ int \ limits _ {2} ^ {n} {\ frac {dt} {\ ln (t)}}}$ ${\ displaystyle \ operatorname {Li} (n) = \ int \ limits _ {2} ^ {n} {\ frac {dt} {\ ln (t)}}}$ - shifted integral logarithm ^[1] .

John Littlewood in 1914 gave unconstructive evidence that such a number exists.

Stanley Skewes in 1933 estimated this number, based on the Riemann hypothesis , as $\exp ^{3}(79)=e^{e^{e^{79}}}\approx 10^{10^{10^{34}}}$ ${\ displaystyle \ exp ^ {3} (79) = e ^ {e ^ {e ^ {79}}} \ approx 10 ^ {10 ^ {10 ^ {34}}}}$ ${\ displaystyle \ exp ^ {3} (79) = e ^ {e ^ {e ^ {79}}} \ approx 10 ^ {10 ^ {10 ^ {34}}}}$ Is the first number of Skuse , denoted by $\mathrm {Sk} _{1}$ ${\ displaystyle \ mathrm {Sk} _ {1}}$ ${\ mathrm {Sk}} _ {1}$ .

In 1955, he also assessed without assuming the validity of the Riemann hypothesis: $\exp ^{4}(7{,}705)=e^{e^{e^{e^{7{,}705}}}}\approx 10^{10^{10^{963}}}$ ${\ displaystyle \ exp ^ {4} (7 {,} 705) = e ^ {e ^ {e ^ {e ^ {7 {,} 705}}}} approx 10 ^ {10 ^ {10 ^ {963 }}}}$ ${\ displaystyle \ exp ^ {4} (7 {,} 705) = e ^ {e ^ {e ^ {e ^ {7 {,} 705}}}} approx 10 ^ {10 ^ {10 ^ {963 }}}}$ - the second number of Skuse , denoted by $\mathrm {Sk} _{2}$ ${\ displaystyle \ mathrm {Sk} _ {2}}$ ${\ mathrm {Sk}} _ {2}$ . This is one of the largest numbers ever used in mathematical proofs, although much less than the Graham number .

In 1987, RJ (HJJ te Riele), without assuming the Riemann hypothesis, limited the Squuse number to $e^{e^{27/4}}$ ${\ displaystyle e ^ {e ^ {27/4}}}$ $e ^ {{e ^ {{27/4}}}}$ which is approximately 8.18510 ³⁷⁰ .

By 2018, it is known that the Skuse number is between 10 ¹⁹ ^[2] and 1.3971672 · 10 ³¹⁶ ≈ e ^{727.951336108} ^[3] .

Notes

↑ Yu. V. Matiyasevich . Alan Turing and Number Theory // Mathematics in Higher Education. - 2012. - No. 10. - S. 111-134.
↑ Jan Büthe. An analytic method for bounding ψ ( x ) // Math. Comp. - 2018 .-- Vol. 87. - P. 1991-2009. - arXiv : 1511.02032 . - DOI : 10.1090 / mcom / 3264 . The proof uses the Riemann hypothesis.
↑ Yannick Saouter, Timothy Trudgian, and Patrick Demichel. A still sharper region where π ( x ) - li ( x ) is positive // Math. Comp. - 2015. - Vol. 84. - P. 2433-2446. - DOI : 10.1090 / S0025-5718-2015-02930-5 . MR : 3356033 . The indicated estimate does not require the Riemann hypothesis.

[1] Yu. V. Matiyasevich . Alan Turing and Number Theory // Mathematics in Higher Education. - 2012. - No. 10. - S. 111-134.

[2] Jan Büthe. An analytic method for bounding ψ ( x ) // Math. Comp. - 2018 .-- Vol. 87. - P. 1991-2009. - arXiv : 1511.02032 . - DOI : 10.1090 / mcom / 3264 . The proof uses the Riemann hypothesis.

[3] Yannick Saouter, Timothy Trudgian, and Patrick Demichel. A still sharper region where π ( x ) - li ( x ) is positive // Math. Comp. - 2015. - Vol. 84. - P. 2433-2446. - DOI : 10.1090 / S0025-5718-2015-02930-5 . MR : 3356033 . The indicated estimate does not require the Riemann hypothesis.

Squuse number

Notes

More articles: