JRM's minuscule probability (~10^-20) reminded me of something that I had just read in a book on the Riemann hypothesis.

The is no 'biggest' number. However, I came across the largest number that I have ever seen used for anything. There are two numbers, Skewes' Number 1 and Skewes' Number 2. These numbers are upper bounds for the size of the first Prime Number for which Gauss's approximation for the number of Prime number up to a given natural number. From Wikipedia: [and note, Graham's number mentioned at the end seems to me to be a different 'type' of number, something that you can essentially make up by specification, but what do I know?]

In number theory, Skewes' number can refer to several extremely large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number x for which

Ï€(x) > li(x),

where Ï€(x) is the prime-counting function and li(x) is the logarithmic integral function. The numbers found by Skewes are now only of historical interest, because computer calculations have produced much smaller estimates. As of 2007[update], these calculations suggest that the smallest such x is close to 1.397×10^316.

John Edensor Littlewood, Skewes' teacher, proved in (Littlewood 1914) that there is such a number (and so, a first such number); and indeed found that the sign of the difference Ï€(x) âˆ’ li(x) changes infinitely often. All numerical evidence then available seemed to suggest that Ï€(x) is always less than li(x), though mathematicians familiar with Riemann's work on the Riemann zeta function would probably have realized that occasional exceptions were likely by the argument given below (and the claim sometimes made that Littlewood's result was a big surprise to experts seems doubtful). Littlewood's proof did not, however exhibit a concrete such number x; it was not an effective result.

Skewes (1933) proved that, assuming that the Riemann hypothesis is true, there exists a number x violating Ï€(x) < li(x) below

e^{e^{e^{79}}}

(now sometimes called first Skewes' number), which is approximately equal to

10^{10^{8.85 \times 10^{33}}}.

In (Skewes 1955), without assuming the Riemann hypothesis, he managed to prove that there must exist a value of x below

10^{10^{10^{963}}}

(sometimes called second Skewes' number). Skewes' task was to make Littlewood's existence proof effective: exhibiting some concrete upper bound for the first sign change. According to George Kreisel, this was at the time not considered obvious even in principle. The approach called unwinding in proof theory looks directly at proofs and their structure to produce bounds. The other way, more often seen in practice in number theory, changes proof structure enough so that absolute constants can be made more explicit.

Skewes' result was celebrated partly because the proof structure used excluded middle, which is not a priori a constructive argument (it divides into two cases, and it is not computable in which case one is working). Although both Skewes numbers are big compared to most numbers encountered in mathematical proofs, neither is anywhere near as big as Graham's number.

The is no 'biggest' number. However, I came across the largest number that I have ever seen used for anything. There are two numbers, Skewes' Number 1 and Skewes' Number 2. These numbers are upper bounds for the size of the first Prime Number for which Gauss's approximation for the number of Prime number up to a given natural number. From Wikipedia: [and note, Graham's number mentioned at the end seems to me to be a different 'type' of number, something that you can essentially make up by specification, but what do I know?]

In number theory, Skewes' number can refer to several extremely large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number x for which

Ï€(x) > li(x),

where Ï€(x) is the prime-counting function and li(x) is the logarithmic integral function. The numbers found by Skewes are now only of historical interest, because computer calculations have produced much smaller estimates. As of 2007[update], these calculations suggest that the smallest such x is close to 1.397×10^316.

John Edensor Littlewood, Skewes' teacher, proved in (Littlewood 1914) that there is such a number (and so, a first such number); and indeed found that the sign of the difference Ï€(x) âˆ’ li(x) changes infinitely often. All numerical evidence then available seemed to suggest that Ï€(x) is always less than li(x), though mathematicians familiar with Riemann's work on the Riemann zeta function would probably have realized that occasional exceptions were likely by the argument given below (and the claim sometimes made that Littlewood's result was a big surprise to experts seems doubtful). Littlewood's proof did not, however exhibit a concrete such number x; it was not an effective result.

Skewes (1933) proved that, assuming that the Riemann hypothesis is true, there exists a number x violating Ï€(x) < li(x) below

e^{e^{e^{79}}}

(now sometimes called first Skewes' number), which is approximately equal to

10^{10^{8.85 \times 10^{33}}}.

In (Skewes 1955), without assuming the Riemann hypothesis, he managed to prove that there must exist a value of x below

10^{10^{10^{963}}}

(sometimes called second Skewes' number). Skewes' task was to make Littlewood's existence proof effective: exhibiting some concrete upper bound for the first sign change. According to George Kreisel, this was at the time not considered obvious even in principle. The approach called unwinding in proof theory looks directly at proofs and their structure to produce bounds. The other way, more often seen in practice in number theory, changes proof structure enough so that absolute constants can be made more explicit.

Skewes' result was celebrated partly because the proof structure used excluded middle, which is not a priori a constructive argument (it divides into two cases, and it is not computable in which case one is working). Although both Skewes numbers are big compared to most numbers encountered in mathematical proofs, neither is anywhere near as big as Graham's number.

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