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"Mathematicians Discover The Ninth Dedekind Number, After 32 Years of Searching

Undeterred after three decades of looking, and with some assistance from a supercomputer, mathematicians have finally discovered a new example of a special integer called a Dedekind number.

Only the ninth of its kind, or D(9), it is calculated to equal 286 386 577 668 298 411 128 469 151 667 598 498 812 366, if you're updating your own records. This 42 digit monster follows the 23-digit D(8) discovered in 1991."

🔗: https://www.sciencealert.com/mathematicians-discover-the-ninth-dedekind-number-after-32-years-of-searching

#DedekindNumber #Dedekind #NumberTheory #Mathematics #Math #science

Mathematicians Christian Jäkel and Lennart Van Hirtum et al. simultaneously discover the 42-digit Dedekind number after 32 years of trying.

The exact values of the Dedekind numbers are known for \(0\leq n\leq9\):
\(2,3,6,20,168,7581,7828354,2414682040998,\)
\(56130437228687557907788,\)
\(286386577668298411128469151667598498812366\)
(sequence A000372 in the OEIS)

🔗 https://scitechdaily.com/elusive-ninth-dedekind-number-discovered-unlocking-a-decades-old-mystery-of-mathematics/?expand_article=1

🔗 https://www.sciencealert.com/mathematicians-discover-the-ninth-dedekind-number-after-32-years-of-searching

Summation formula👇
Kisielewicz (1988) rewrote the logical definition of antichains into the following arithmetic formula for the Dedekind numbers:
\[\displaystyle M(n)=\sum_{k=1}^{2^{2^n}} \prod_{j=1}^{2^n-1} \prod_{i=0}^{j-1} \left(1-b_i^k b_j^k\prod_{m=0}^{\log_2 i} (1-b_m^i+b_m^i b_m^j)\right)\]

where \(b_i^k\) is the \(i\)th bit of the number \(k\), which can be written using the floor function as
\[\displaystyle b_i^k=\left\lfloor\frac{k}{2^i}\right\rfloor - 2\left\lfloor\frac{k}{2^{i+1}}\right\rfloor.\]

However, this formula is not helpful for computing the values of \(M(n)\) for large \(n\) due to the large number of terms in the summation.

Asymptotics:
The logarithm of the Dedekind numbers can be estimated accurately via the bounds
\[\displaystyle{n\choose \lfloor n/2\rfloor}\le \log_2 M(n)\le {n\choose \lfloor n/2\rfloor}\left(1+O\left(\frac{\log n}{n}\right)\right).\]

Here the left inequality counts the number of antichains in which each set has exactly \(\lfloor n/2\rfloor\) elements, and the right inequality was proven by Kleitman & Markowsky (1975).

#DedekindNumber #Dedekind #NumberTheory #Mathematics #Sequence #Discovery #Mathematicians #Challenging #RichardDedekind #DifficultProblem #MathHistory #Pustam #ChallengingProblem #EGR #PustamRaut