7 votes
Accepted

Function identifying consecutive integers

It's possible to make a polynomial with this property. If $p$ is a polynomial with real coefficients, let $Z(p)$ denote the set of real zeros of $p$. Then, for polynomials $p$ and $q$, $$Z(p^2+q^2)=Z(...
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5 votes
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Find real number $x$ such that both $x+\sqrt{2022}$ and $\frac{3}{x}-\sqrt{2022}$ is an integer

We know that $2022$ doesn't contain a square (as $2022=2*3*337$). So the solution must be of the form $x=n-\sqrt{2022}$ to satisfy the first equation, where $n\in \mathbb Z$. Injection in the 2nd ...
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  • 1,057
3 votes
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Proving $\mathbb{Z}$ is a normal subgroup of $\mathbb{Z}[i]$

Lulu is right in their comment - in an abelian group, every subgroup is normal. In fact you prove this precisely in your Part 2 - notice how you only use the fact that addition is commutative. Part 1 ...
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3 votes

Function identifying consecutive integers

One particularly simple polynomial is $$f(a,b,c)=a^2+b^2+c^2-ab-bc-ca-3,$$ whose only integer zeros are when $\{a,b,c\}=\{n-1,n,n+1\}$ for some $n$, but which has other real zeros. To show that $f(a,b,...
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2 votes

Is the structure of sequence A094615 obvious?

That linked page has an explicit representation of the entries given by Lamine Ngom, namely $$T(n,k) = 2^{n+1-k}3^{k-1} -1 $$ Should be not too hard to verify by induction that the triangular pattern ...
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2 votes

Proof help for the existence of a function in Velleman 3rd Chapter 8 section 2

Define $f$ recursively as follows. First set $f(1)=1$ and set $N_1 = 1$, $N_0 = 0$. Now, suppose that for some $n\geq 1$, $f(l)$ has been defined for all $l\leq N_n$. Let $N_{n+1} = N_{n+1} + (N_{n+1}-...
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1 vote
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The explicit monoid law on the differences of two square integers?

$(a^2+b^2)(c^2+d^2)=(ac+bd)^2+(ad-bc)^2$ is a famous identity. Now substitute $b=ib’,d=id’$ to obtain $$(a^2-b^2)(c^2-d^2)=(ac-bd)^2-(ad-bc)^2$$.
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  • 2,096
1 vote

Proof help for the existence of a function in Velleman 3rd Chapter 8 section 2

Think of listing the possible triples $(a, b, c)$ in order (it's a countable set). Now sequentially choose an $n(a, b, c)$ for each and define $f(a \cdot n(a, b, c)+b) = c$. You need to make sure that ...
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  • 81
1 vote
Accepted

Compressing the primes using simple addition?

Here's what I've found. I was concentrating more on keeping $\sup(A+B)$ fairly small than on maximising $|A+B|$. \begin{array}{rrllrr} |A|&|B|&A&B&\sup(A+B)&|A+B|\\ 5&5& 5, ...
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  • 2,463

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