The Masked Avenger
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Member for 11 years, 5 months
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Last seen more than 9 years ago
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Bounds for constructing $n!$ with additions, subtractions, and multiplications starting from $1$
There may also be a computation string which consists solely of all the primes and their powers, possibly in increasing order.
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Bounds for constructing $n!$ with additions, subtractions, and multiplications starting from $1$
You can also generate primes by generating the gaps first. For k less than 18 decimal digits this is less than O((log k)^2) .
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Sets of points containing permutations - a Ramsey-type question
Interesting. I wonder how low (k+l) can go, using the notation in Paseman's comment.
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Enumerating matrices function of ranks
See Zivkovic's 2005 arxiv print on classification of small binary matrices. The table inside goes up to n=8, I believe.
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Bounds for constructing $n!$ with additions, subtractions, and multiplications starting from $1$
1,2,3,6,36,72,70, so t(5040) is at most 8. Have not found a shorter string yet.
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Sets of points containing permutations - a Ramsey-type question
It breaks down at n=2, with a coloring in which l is at most 1 and k at most 2. Take the main and lower diagonals as 7 elements of one color.
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Sets of points containing permutations - a Ramsey-type question
Also I don't see the first easy observation: can you provide an example where B has more than half the nodes and not all the permutations of n/2?
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Sets of points containing permutations - a Ramsey-type question
I don't know if this precise form is in the literature. Looking at permanents, binary matrices, and enumerating matrices avoiding a certain pattern may get you literature which gets close to the form above.
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Analogues of Primitive Recursive Functions
When I read this I am reminded of Fenstad and Abstract Recursion Theory. I don't quite know why; it may not fit in with your program.
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Common sizes of intersections
@Tony. Indeed. I withdraw the suggestion of "adding A".
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How did Ramanujan discover this identity?
She did it by plugging in values for the variables. That, or she consulted an oracle who performed the infinitely many computations.
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Common sizes of intersections
Huh. I meant singletons. I guess I should have said it. Oh well, just add A the whole set to Tony's example.
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Common sizes of intersections
Why not add A to it for a slightly larger set? Not to mention the empty set.
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Covering a set with images of a transversal
Just to be clear: each omega_i is a G-set? Or is something more wild going on?
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Conjecture on the square root of the sum of the squares of the prime factors of a number
So the last statement is that I believe in the existence of an n such that for i from 0 to 4, n+i is composite and A_(n+i) is integral. I further believe (but cannot prove) that such an n is greater than 10^16.
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Conjecture on the square root of the sum of the squares of the prime factors of a number
I'm sorry for being unclear. I'm with Greg in thinking there are arbitrarily long sequences of integers n which are intervals of integers with no primes, and all of whose A values are integral. I would be surprised to find an interval of 5 such numbers below 10^20.
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Conjecture on the square root of the sum of the squares of the prime factors of a number
Further, any composite n for which A_n is integral must have a prime factor smaller than the fourth root of n. Thus you can stop your trial division if the smallest factor of n is greater than n^1/4. For small ranges, this means you can use certain sieve techniques to replace your factoring step.
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Conjecture on the square root of the sum of the squares of the prime factors of a number
No, I am counting repeats. Squares are 0 or 1 mod 4, so if the sum of squares is a square, there can't be (say) exactly 6 odd squares in the sum. Regarding bitlength, if n=pq with p large enough, q^2 smaller than 2p, p^2 + q^2 will be smaller than (p+1)^2, and again A_n will not be integral.
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Conjecture on the square root of the sum of the squares of the prime factors of a number
Something that might speed up your search: A_n cannot be an integer if n has (2 or 3 mod 4)-many odd prime factors, nor can it be integral if n is composite with largest prime factor p having bitlength more than 2/3 the bitlength of n. These tests may be faster than doing a square root of a sum of squares.