Is every positive integer the rank of an elliptic curve over some number field?

For every positive integer $$n$$, is there some number field $$K$$ and elliptic curve $$E/K$$ such that $$E(K)$$ has rank $$n$$?

It’s easy to show that the set of such $$n$$ is unbounded. But can one show that every positive integer is the rank of some elliptic curve over a number field?

The analogous question for a fixed number field is expected to have a negative answer (c.f. e.g., this question) but is still conjectural. But I wonder if one might be able to prove a positive answer to the question I asked above.

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Finding rank of newly inserted node in AVL tree augmented with rank of node

My prof was talking about how to augment a tree to efficiently find the key with a given rank. On the way to getting the right answer (store the size of the subtree rooted at each node), he proposed augmenting each node with it’s rank. That way finding the node with a certain rank is a simple binary search. But he said this wouldn’t work because insertion would be $$\Theta(n)$$. I understand that this is because in the worst case the newly inserted node has rank 0 and so the rank of each node in the tree must be incremented by 1. What I am interested in knowing is an algorithm for computing the rank of a newly inserted node. At first I thought that if a node $$n$$ is inserted to the left of a leaf $$l$$ with rank $$k$$ then the rank of $$n$$ is $$k-1$$, or $$k+1$$ if inserted right of $$l$$. I found a counter example to that so I know it is not the case. So how would this be done?

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Is my brute force / naive solution of the given Hacker Rank problem correct?

Just help me with the brute force approach only.

Problem Name: Game of Two Stacks Problem Link

Note: I have converted vector to stack for twoStacks() function.

Problem Description: Alexa has two stacks of non-negative integers, stack A and stack B where index 0 denotes the top of the stack. Alexa challenges Nick to play the following game:

In each move, Nick can remove one integer from the top of either stack A or B stack.

Nick keeps a running sum of the integers he removes from the two stacks.

Nick is disqualified from the game if, at any point, his running sum becomes greater than some integer X given at the beginning of the game.

Nick’s final score is the total number of integers he has removed from the two stacks.

find the maximum possible score Nick can achieve (i.e., the maximum number of integers he can remove without being disqualified) during each game and print it on a new line.

For each of the games, print an integer on a new line denoting the maximum possible score Nick can achieve without being disqualified.

My Solution:

int twoStacks(int x, stack<int> a, stack<int> b) {     if(x < 0) {         return -1;     }     int a_top = -1, b_top = -1, m1 = 0, m2 = 0;     if(!a.empty()) {    // Pop A         a_top = a.top();         a.pop();         m1 = 1 + twoStacks(x - a_top, a, b);     }     if(!b.empty()) {    // Pop B         b_top = b.top();         b.pop();         if(a_top != -1) {             a.push(a_top);         }         m2 = 1 + twoStacks(x - b_top, a, b);     }     return max(m1, m2); } 
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