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Difference between revisions of "User:Ddk001"

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See if you can solve these:  
 
See if you can solve these:  
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1. There is one and only one perfect square in the form
 
1. There is one and only one perfect square in the form
  

Revision as of 17:47, 1 January 2024

See if you can solve these:

1. There is one and only one perfect square in the form

$(p^2+1)(q^2+1)-((pq)^2-pq+1)$

Find that perfect square.

2. Suppose there is complex values $x_1, x_2,$ and $x_3$ that satisfy

$(x_i-\sqrt[3]{13})((x_i-\sqrt[3]{53})(x_i-\sqrt[3]{103})=\frac{1}{3}$

Find $x_{1}^3+x_{2}^3+x_{2}^3$.

3. Suppose

$x \equiv 2^4 \cdot 3^4 \cdot 7^4+2^7 \cdot 3^7 \cdot 5^6 \pmod{7!}$

Find the remainder when $\min{x}$ is divided by 1000.

4. Suppose $f(x)$ is a $10000000010$-degrees polynomial. The Fundamental Theorem of Algebra tells us that there are $10000000010$ roots, say $r_1, r_2, \dots, r_{10000000010}$. Suppose all integers $n$ ranging from $-1$ to $10000000008$ satisfies $f(n)=n$. Also, suppose that

$(2+r_1)(2+r_2) \dots (2+r_{10000000010})=m!$

for an integer $m$. If $p$ is the minimum possible value of

$(1+r_1)(1+r_2) \dots (1+r_{10000000010})$.

Find the number of factors of the prime $999999937$ in $p$.

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