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Graham, Ronald L.
Termination of the sixth Goodstein Sequence ★
Author(s): Graham
Question How many steps does it take the sixth Goodstein sequence to terminate?
Keywords: Goodstein Sequence
Monotone 4-term Arithmetic Progressions ★★
Author(s): Davis; Entringer; Graham; Simmons
Question Is it true that every permutation of positive integers must contain monotone 4-term arithmetic progressions?
Keywords: monotone arithmetic progression; permutation
Pebbling a cartesian product ★★★
Author(s): Graham
We let denote the pebbling number of a graph
.
Conjecture
.
![$ p(G_1 \Box G_2) \le p(G_1) p(G_2) $](/files/tex/3371d3b2280f593cc1fc3a0b3da7a685c3525910.png)
Divisibility of central binomial coefficients ★★
Author(s): Graham
Problem (1) Prove that there exist infinitely many positive integers
such that
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$$\gcd({2n\choose n}, 3\cdot 5\cdot 7) = 1.$$](/files/tex/9a2e7bf7e3a51324285eef8a82042a439d4c15b6.png)
Problem (2) Prove that there exists only a finite number of positive integers
such that
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$$\gcd({2n\choose n}, 3\cdot 5\cdot 7\cdot 11) = 1.$$](/files/tex/57b37bf3adc8403140096ce71611b82e09ece96f.png)
Keywords:
The large sets conjecture ★★★
Author(s): Brown; Graham; Landman
Conjecture If
is 2-large, then
is large.
![$ A $](/files/tex/7a8d9782350e8eb5a84c149576d83160492cbdd3.png)
![$ A $](/files/tex/7a8d9782350e8eb5a84c149576d83160492cbdd3.png)
Keywords: 2-large sets; large sets
Graham's conjecture on tree reconstruction ★★
Author(s): Graham
Problem for every graph
, we let
denote the line graph of
. Given that
is a tree, can we determine it from the integer sequence
?
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ L(G) $](/files/tex/76f6ee75811ed6d89f7e99f8aa7a505f462c30b2.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ |V(G)|, |V(L(G))|, |V(L(L(G)))|, \ldots $](/files/tex/b93a7a2773f22770891c213297401ce7189f4fa2.png)
Keywords: reconstruction; tree
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