Maximal Multiplication of All Possible Summands
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8/9/2019 Maximal Multiplication of All Possible Summands
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n ∈ N n = x1 + ... + xk
k xi
k xi
n
n
8/9/2019 Maximal Multiplication of All Possible Summands
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1 ≤ k ≤ n
Rec(iter, num)
Rec(k, n)
num
1 ≤ a ≤ n Rec(iter − 1,num − a)
n k
π(x1,...,xk) = x1 · ... · xk
xi = 0
x1 + ... + xk = n
π(x1,...,xk−1) = x1 · ... · xk−1(n − x1 − ... − xk−1)
∂π
∂xi
= j=i
xj · (n − x1 − ... − xk−1 − xi)
π = −→
0
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2 1 ... 11 2 ... 1
... ... ... ...1 1 ... 2
x1x2
...xk−1
=
n
n
...n
xi = nk
x1 ... xk
=
nk
... nk
π(x1,...,xk) = x1 · ... · xk
π(k) = (n
k)k
k ≥ 1
dπ
dk = (
n
k)k(log((
n
k) − 1)
n
k = e k = n
e n ≥ 3
n
e + e + ... + e
ne
en
e
N
(n1,...,nk) (e,...,e)
3 + ... + 3
n3
3 + ... + 3 + 2
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3 + ... + 3 + 1 3 + ... + 3 + 2 + 2 (3 − e)2 + (e − 1)2 > 2(e − 2)2
n
3 + ... + 3 3|n
3 + ... + 3 + 2 + 2 3|n − 1
3 + ... + 3 + 2 3|n − 2
3n
3 20 3|n
3n−1
3 −122 3|n − 1
3n−2
3 21 3|n − 2
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