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1 vote

### Is there any proof that there are no integers, between 0 and 1 (that doesn´t use the well-ordering principle)?

There cannot be such a proof because other rings like $\mathbb{Q}$ satisfy the same algebraic and order axioms, but do have an element $x$ with $0 < x < 1$. The well ordering principle is what ...
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• 1,749
1 vote

### Finding the sum of the floor function of $a,(b-1)/2,c$ given two symmetric sums

how would you solve it? Suppose that two of $a,b,c$ are integers. Then, it follows from $a+b+c=6$ that $a,b,c$ are integers. We have $abc\not=0$. (If $x+y=6$ and $xy=9$, then $x,y$ are the solutions ...
• 145k

• 1,032
Accepted

### Are there infinitely many Munchausen numbers?

Let's say a Munchausen number has $d$ digits (with non-zero leading digit). On one hand, the number is at least $10^{d-1}$. On the other hand, the number is at most $9^9d$ by the Munchausen property. ...
• 10.7k

### Finding all $x+y+z$ for positive integers satisfying $\frac{13}{x^2}+\frac{28}{y^2}=\frac{z}{85}$

This doesn't looks short, but here's a hint that reduces the problem to a finite search: Write the equation as $(x^2z-1105)(y^2z-2380)=2629900$. This means that both $x^2z-1105$ and $y^2z-2380$ are ...
• 19.1k
1 vote

1 vote

• 10.9k
1 vote

### Is every integer a quadratic residue mod some p?

A trivial answer is "yes, modulo $p=2$, every integer is a quadratic residue".
• 133k

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