GingerAle

Actually, HSC, the reason you didn’t need to do the reCAPTCHA was because you reached the 1000 point level. 

The reCAPTCHA thing always annoyed Naus, especially if he was on a finicky network.  The funny thing was he would have had well more than a 1000 points if it weren’t for all the negative points he received (back when you could get them)—he thought of negative points as high marks of excellence for his trolling. It didn’t matter anyway, he could have given himself as many points as he wanted –he never would though.  Naus hasn’t been on for almost a year now, and his count jumped over 600 points with the new update.  

Wow! We are having a meltdown, aren’t we? What happened? Did you lose your canteen privileges?

My answer is the same as Sir Alan’s?  Did you use a computer to analyze this or did you do it the old-fashioned way?  Never mind, I already know the answer.  Of course, my answer is the same as Sir Alan’s, because brilliant minds think alike. Our answers are both correct, so they have to be the same.

I was trying to figure out how long it would take to type them in Roman numerals.  A long time, but not as long as Sir CPhill’s quest. 

That number is the largest currently known Mersenne Prime.

Corrected error (and clarified math for the hoary hair tortuous).

$\text {(a)}\\ \small v_f \small \text { velocity of father. } v_s \text { velocity of son. } \\ \frac{1}{2} m_f(v_f + 1.0)^2 = (2)* \frac{1}{2} m_fv^2_f \\ (v_f + 1.0)^2 = 2v^2_f \hspace{2em} \small \text {| Divide by } m_f \text { and reduce}\\ v_f +1 = \sqrt{2}v_f \hspace{3em} \text { | square root of both sides}\\ v_f + 1 \approx 1.4142v_f \\ 0.4142v_f \approx 1 \hspace{4em} \text{ | subtract and divide}\\ v_f \approx 2.4142m/s \\$

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$\text {(b)}\\ \text { } \\ \frac{1}{2}(m_f/2)v_s^2 = (2) * \frac{1}{2} (m_fv^2_f) \\ \frac {1}{4}v_s^2 = v^2_f \hspace{3em} \small \text {|Divide by } m_f \text { and reduce}\\ v_s^2 = 4v_f^2 \hspace{3em} \small \text { |Multiply through by 4)}\\ v_s = 2v_f \hspace{3em} \small \text { |Take square root }\\ v_s = 2*(2.4142) \approx 4.8284 m/s$

Sometimes I need a second application of the Tortoise & Hair dye AI formula – especially when the blarney banker is nearby.  If you had used it, then this wouldn’t have been a "W T F" meltdown moment –it would be just a mistake.

Solution for original velocities of father & son, via kinetic energy

$\text {(a)}\\ \small v_f \small \text { velocity of father. } v_s \text { velocity of son. } \\ \frac{1}{2} m_f(v_f + 1.0)^2 = 2 × ½ m_fv^2_f \\ (vf + 1.0)^2 = 2v^2_f \\ v_f +1 = \sqrt{2}v_f \hspace{3em} \text { | square root of both sides}\\ v_f + 1 \approx 1.4142v_f \\ v_f \approx 2.4143v_f \\ 0.4142v_f \approx 1 \\ v_f \approx 1.4143 \\ \text { } \\ \text {(b)}\\ \frac{1}{2}v_s^2 = 2 * \frac{1}{2} (m_fv^2_f) \\ \frac {1}{4}v_s^2 = v^2_f \\ v_s^2 = 4v_f^2\\ v_s = 2v_f \approx 4.8286 m/s$

$\small \text{Theory & Formulas: Complements of Gottfried Wilhelm Leibniz, with Christiaan Huygens & René Descartes. }\\ \small \text{ }\hspace{17em}\scriptsize \text {(Amazingly, there is very little of Sir Isaac Newton in this.) }\\ \small \text{Produced by Lancelot Link and company. }\\ \small \text{Directed by GingerAle. }\\ \small \text{Sponsored by Naus Corp. Tortoise and Hair Dye: }\\ \small \text{ }\hspace{9em} \small \text {Artificial intelligence formula for slow blondes and some mathematicians. }\\ \small \text{ }\hspace{9em} \small \text {New formula now works for red-heads. } \scriptsize \text{(Helps to prevent communist sympathies, too.) }$

Solution for (n) (smallest positive integer that satisfies the system of congruencies).

n mod 1103 = 1041

n mod 1303 = 859

n mod 2003 = 1095

$\begin{array}{rcll} n &\equiv& {\color{red}1041} \pmod {{\color{green}1103}} \\ n &\equiv& {\color{red}859} \pmod {{\color{green}1303}} \\n &\equiv& {\color{red}1095} \pmod {{\color{green}2003}} \\ \text{Let } m &=&1103 \cdot 1303\cdot 2003 = 2878729627 \\ \end{array}$

$\text {1103, 1303, and 2003 are coprime numbers (they are actually prime).}\\$

$\small{ \begin{array}{l} n = {\color{red}1041} \cdot {\color{green}1303\cdot 2003} \cdot \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ [ { (\color{green}1303\cdot 2003)}^{\varphi({\color{green}1103}) -1 } \pmod {{\color{green}1103}} ] }_{=\text{modulo inverse }(1303\cdot 2003) mod 1103 }}_{=(1303\cdot 2003)^{1103-1} \mod {1103}} }_{=(1303\cdot 2003)^{1102} \mod {1103}} }_{=(2609909\pmod{1103})^{1102} \mod {1103}} }_{=(211)^{1102} \mod {1103}} }_{=988} + {\color{red}859} \cdot {\color{green}1103\cdot 2003} \cdot \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ [ { (\color{green}1103\cdot 2003) }^{\varphi({\color{green}1303}) -1} \pmod {{\color{green}1303}} ] }_{=\text{modulo inverse } (1103\cdot 2003) mod 1303 } }_{=(1103\cdot 2003)^{1302-1} \mod {1303}} }_{=(1103\cdot 2003)^{1301} \mod {1303}} }_{=(2209309\pmod{1303})^{1301} \mod {1303}} }_{=(724)^{1301} \mod {1303}} }_{=9} +{\color{red}{1095}} \cdot {\color{green}1103\cdot 1303} \cdot \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ [ { (\color{green}1103\cdot 1303) }^{\varphi({\color{green}2003}) -1 } \pmod {{\color{green}2003}} ] }_{=\text{modulo inverse } (1103\cdot 1303) mod 2003 } }_{=(1103\cdot 1303)^{2002-1} \mod {2003}} }_{=(1103\cdot 1303)^{2001} \mod {2003}} }_{=(1437209\pmod{2003})^{2001} \mod {2003}} }_{=(1058)^{2001} \mod {2003}} }_{=195}\\\\ n = {\color{red}{1041}} \cdot {\color{green}{1303}\cdot 2003} \cdot [988] + {\color{red}859} \cdot {\color{green}1103\cdot 2003} \cdot [9] + {\color{red}1095} \cdot {\color{green}1103\cdot 1303} \cdot [195] \\ n = 2684312285772 + 17080167879 + 306880051725 \\ n = 3008272505376 \\\\ n \pmod {m}\\ = 3008272505376 \pmod {2878729627} \\ = 45161 \\\\ n = 45161 + k\cdot 2878729627 \qquad k \in Z\\\\ \mathbf{n_{min}} \mathbf{=} \mathbf{45161} \end{array}}$

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$\small \text {Related formulas and principles compliments of Leonhard Euler }\scriptsize \text {(totient function),} \\ \small \text {Euclid of Alexandria }\scriptsize \text {(Extended Euclidean algorithm), and Brilliant Chinese mathematicians “Chinese Remainder Theorem” } \\ \small \text {LaTex layout and coding adapted from Heureak’s mathematical solution and Latex presentation:}\\ \tiny \text {http://web2.0calc.com/questions/find-the-smallest-positive-integer-that-satisfies-the-system-of-congruences }\\ \small \text {Produced by Lancelot Link & Co.}\\ \small \text {Directed by GingerAle}\\ \small \text {Sponsored by Nause Corp pharmaceuticals: Makers of } \\ \scriptsize \text { Quantum Vaccines for spooky dumbness at a distance and related contagious dumbness diseases (rCDDs).}\\ \scriptsize \text{ and}\\ \scriptsize \text{ Master Blarney Filters. Now filters most toxins emitted by blarney bankers and related dumb-dumbs. } \\ $

That’s true. If he wasn’t a cute duck then he would cross at more than 90° and no one would say why because that would be obtuse.  

You know, I am often amazed how many pets seem to like the friendly neighbor more than they do their master or mistress. 

For over twelve years, I’ve volunteered at a pet shelter, and we’ve had about every kind of pet you might think of: Along with dogs, cat and birds, there have been many reptiles (including an alligator), snakes (including the poisonous kind) and even monkeys (never a chimp, but one works there, though), but one thing we’ve never had was a pet rock. I should bring one in.

Happy (belated) Birthday, Sir Alan.

Our troll, Lancelot Link, told me, “The world and certainly this forum would be notably dumber if Sir Alan were never born.”   

I'm sure that is true.