MRB constant rational-denominator interval certificate
Generated from uploaded file: 6p5millionMRB.txt
Generated from uploaded file: 6p5millionMRB.txt
A. Published 6,500,000-digit certificate
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Interpret the first N = 6,500,000 digits after the decimal point as a truncation interval
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Interpret the first N = 6,500,000 digits after the decimal point as a truncation interval
A/10^N <= C_MRB < (A+1)/10^N.
SHA-256 of the N decimal digits (digits only, no leading "0."):
3942ea7058b90b13ce5f08215657c3b279dcec950ec822963c45a202cf4acac0
3942ea7058b90b13ce5f08215657c3b279dcec950ec822963c45a202cf4acac0
The rational with smallest positive denominator in this interval is p/q,
where p and q are stored exactly in:
mrb_min_fraction_6p5m.txt
(first line p, second line q).
where p and q are stored exactly in:
mrb_min_fraction_6p5m.txt
(first line p, second line q).
p has 3,250,000 decimal digits.
q has 3,250,001 decimal digits.
q has 3,250,001 decimal digits.
q begins:
10455575851459938617474587027070078078405782337740854067259840612936276203995851
q ends:
08998801674603789561807788844175808943315162223085962544843959015505443237485524
10455575851459938617474587027070078078405782337740854067259840612936276203995851
q ends:
08998801674603789561807788844175808943315162223085962544843959015505443237485524
Thus
q = 1.0455575851459938617474587027070078078405782337740854... x 10^3,250,000.
q = 1.0455575851459938617474587027070078078405782337740854... x 10^3,250,000.
Exact verification:
gcd(p,q) = 1
A*q <= p*10^N < (A+1)*q
gcd(p,q) = 1
A*q <= p*10^N < (A+1)*q
By the continued-fraction / Stern-Brocot simplest-fraction theorem,
no rational in the 6,500,000-digit interval has denominator smaller than q.
Therefore, if C_MRB is rational and the first 6,500,000 digits are correct,
no rational in the 6,500,000-digit interval has denominator smaller than q.
Therefore, if C_MRB is rational and the first 6,500,000 digits are correct,
denominator(C_MRB) >= q.
The uploaded file contains additional digits, and its very next supplied digit
after the 6,500,000-digit prefix differs from the extremal p/q candidate.
Hence, if that next supplied digit is also trusted,
after the 6,500,000-digit prefix differs from the extremal p/q candidate.
Hence, if that next supplied digit is also trusted,
denominator(C_MRB) > q.
B. Stronger bound using 6,532,400 supplied digits
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For an additional computation, N = 6,532,400 supplied digits were used.
This leaves a guard below the precision annotation present in the uploaded Mathematica output.
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For an additional computation, N = 6,532,400 supplied digits were used.
This leaves a guard below the precision annotation present in the uploaded Mathematica output.
SHA-256 of these N decimal digits:
b4e82442239208f8b9c172b4b4af9bf181cd9b59bc5d8b0676920a34fcd1bc18
b4e82442239208f8b9c172b4b4af9bf181cd9b59bc5d8b0676920a34fcd1bc18
The smallest-denominator fraction in this narrower interval is stored in:
mrb_min_fraction_6532400.txt
mrb_min_fraction_6532400.txt
Its denominator has 3,266,200 decimal digits and begins
23826968384056407318154999138713577093445319256968010080728332607226135623902325
and ends
50620696344284794934188107395585774858837136795525393674205464832893910209003992
23826968384056407318154999138713577093445319256968010080728332607226135623902325
and ends
50620696344284794934188107395585774858837136795525393674205464832893910209003992
Thus its scale is
2.3826968384056407318154999138713577093445319256968010... x 10^3,266,199.
2.3826968384056407318154999138713577093445319256968010... x 10^3,266,199.
Exact verification:
gcd(p,q) = 1
A*q <= p*10^N < (A+1)*q
gcd(p,q) = 1
A*q <= p*10^N < (A+1)*q
The next supplied digit also differs from this extremal rational candidate,
so, if all 6,532,401 digits used in that comparison are trusted, the actual
rational denominator would have to be strictly larger than this q as well.
so, if all 6,532,401 digits used in that comparison are trusted, the actual
rational denominator would have to be strictly larger than this q as well.
Method
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The smallest-denominator rational in an interval is found from the common
continued-fraction prefix of the interval endpoints. At the first divergent
partial quotient, the corresponding Stern-Brocot mediant is the unique
fraction of smallest denominator in the interval. The computation was done
with exact GMP integer arithmetic. A half-GCD implementation was used only
to accelerate the continued-fraction matrix calculation; every returned
matrix was checked against both exact interval endpoints before acceptance
------
The smallest-denominator rational in an interval is found from the common
continued-fraction prefix of the interval endpoints. At the first divergent
partial quotient, the corresponding Stern-Brocot mediant is the unique
fraction of smallest denominator in the interval. The computation was done
with exact GMP integer arithmetic. A half-GCD implementation was used only
to accelerate the continued-fraction matrix calculation; every returned
matrix was checked against both exact interval endpoints before acceptance