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SuicideFuel Math thread problem (official)

Ahnfeltia

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There is no creativity involved in the calculation of values.
I beg to differ. Regarding the problem whose solution you quoted, it's anything but obvious that any n > 32 cannot yield a valid solution, yet all it took to prove this highly nonobvious claim was high school-level math. Moreover, one cannot simply exhaust all those infinitely many options, so some sort of trick is required. How is finding a way to make the seemingly impossible possible with only simple tools not creative?
 
Ahnfeltia

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Since a, b, and c are coprime by the definition of a primitive Pythagorean triple, WLOG at the very least the only prime divisor of a is 2, b is 3, and c is 5. Every number has a prime divisor btw. Therefore a*b*c=2*3*5=30. Hence rad(abc)=30

Did I get it right :feelsaww:
Your solution is unfortunately flawed. Here are a couple of things you overlooked:
  • Just because only one of a, b, and c can have 2 as a prime divisor doesn't necessarily mean that one of them does. They could potentially all be odd. Similarly for 3 and 5. This oversight is the main flaw of your argument. It's incidentally also the bulk of the work.
  • Technically not every positive integer has a prime divisor. 1 does not have any.
  • Once one has properly deduced that WLOG 2 divides a, 3 divides b, and 5 divides c, then one merely knows that 30 divides abc, whence rad(abc) is at least 30. Your equal signs are unwarranted. In fact, there's no Pythagorean triple wherefor abc = 30.
 
Orzmund

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How is finding a way to make the seemingly impossible possible with only simple tools not creative?
The operation and character of values have already been determined via the innate laws of mathematics, particularly those pertaining to the arithmetic and geometry. Algebraic reasoning does discredit this, you could provide a method of proving 2+2=5, but that does not prove that 2+2≠4.

If we eliminate Euclid's fifth postulate one might hastily conclude that they have stumbled upon impossible shapes, yet these shapes were never impossible, they simply remained undiscovered by man for a great deal of time.


so some sort of trick is required
Exactly.
 
Orzmund

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For those who are curious, here's an excellent exposition

I remembered seeing this video a long time ago (I just rewatched it)

Is there an introduction for complex analysis, calculus, and the varying other forms of mathematics that would at least be comprehensive to someone who only has a high-school education? I have books on complex analysis, problem solving, and other topics but they are all too advanced despite having been advertised for beginners.
 
Orzmund

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Let (a,b,c) be a primitive Pythagorean triple. Prove that rad(abc) is at least 30, where rad(n) denotes the radical of the positive integer n -- i.e., the product of the distinct prime divisors of n (with rad(1) = 1).
What does that even mean?

Is every number of n{1,2,3,4,...} being divided by a different number?

Are the three integers representing each side of a supposed right triangle prime numbers? Is that why it is called a "primitive" Pythagorean triple?
Bonus (difficult): prove that the only primitive Pythagorean triple for which rad(abc) = 30 is (a,b,c) = (2,3,5).
 
Ahnfeltia

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you could provide a method of proving 2+2=5, but that does not prove that 2+2≠4.
Please do. I'm skeptical.
If we eliminate Euclid's fifth postulate one might hastily conclude that they have stumbled upon impossible shapes, yet these shapes were never impossible, they simply remained undiscovered by man for a great deal of time.
Who is the "one" you speak of here? I suppose hypothetically one could, but could hypothetically also stick a rod up his ass. I fail to see the relevance tbh.
Is there an introduction for complex analysis, calculus, and the varying other forms of mathematics that would at least be comprehensive to someone who only has a high-school education? I have books on complex analysis, problem solving, and other topics but they are all too advanced despite having been advertised for beginners.
Calculus is meant to follow up on high school math, so most introductory calculus textbooks should suffice. Complex analysis is quite a bit more advanced and requires more mathematical maturity. Most complex analysis books presume familiarity with real analysis at the very least.
What does that even mean?

Is every number of n{1,2,3,4,...} being divided by a different number?
Sorry not sure what you're asking here.
Are the three integers representing each side of a supposed right triangle prime numbers? Is that why it is called a "primitive" Pythagorean triple?
No, in a primitive Pythagorean triple no two of the three integral side lenghts are to share any prime factors. That way any Pythagorean triple which is not primitive has all three sides sharing a common factor and can therefore be downscaled (as a triangle) so as to obtain a primitive one. Primitive Pythagorean triples cannot be downscaled, however, making them somehow fundamental.
 
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Orzmund

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Please do. I'm skeptical.

There was an attempt.

2 + 2 ≠ 2 + 2 if 2 + 2 = 4
Who is the "one" you speak of here? I suppose hypothetically one could, but could hypothetically also stick a rod up his ass. I fail to see the relevance tbh.

Calculus is meant to follow up on high school math, so most introductory calculus textbooks should suffice. Complex analysis is quite a bit more advanced and requires more mathematical maturity. Most complex analysis books presume familiarity with real analysis at the very least.

Sorry not sure what you're asking here.

No, in a primitive Pythagorean triple no two of the three integral side lenghts are to share any prime factors. That way any Pythagorean triple which is not primitive can be downscaled so as to obtain a primitive one. Primitive Pythagorean triples cannot be downscaled, however, making them somehow fundamental.
Fundamental or paradoxical? Your description is vague.

What do you mean by "downscaled"? Decreased in value(subtracted) or factored?
 
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Ahnfeltia

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https://www.quora.com/How-can-we-make-2+2-5-1
There was an attempt.

2 + 2 ≠ 2 + 2 if 2 + 2 = 4
Yeah, I'm still not convinced.
What do you mean by "downscaled"? Decreased in value(subtracted) or factored?
Sorry for being vague. I meant that every triangle corresponding to a Pythagorean triple which is not primitive is similar to a triangle corresponding to a primitive Pythagorean triple. In this case the triangle corresponding to the primitive Pythagorean triple will always be smaller, meaning it's like a downscaled version. No two different primitive Pythagorean triples have similar corresponding triangles.
 
I

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Your solution is unfortunately flawed. Here are a couple of things you overlooked:
  • Just because only one of a, b, and c can have 2 as a prime divisor doesn't necessarily mean that one of them does. They could potentially all be odd. Similarly for 3 and 5. This oversight is the main flaw of your argument. It's incidentally also the bulk of the work.
I wasn't necessarily trying to say that one of them necessarily had 2 as a prime divisor, I'm saying the smallest product of 3 distinct prime numbers is 30 which means that the radical of abc would have to be at least 30. Since a, b, and c are coprime then each would have to have a distinct prime factor. If you try another primitive Pythagorean triple then you would have to change at least one of those prime factors to be a number greater than 2, 3, or 5, which has to have a product greater than 30.

  • Technically not every positive integer has a prime divisor. 1 does not have any.
1 is the only exception. I guess you can exhaustively prove that 1 is not in a primitive pythagorean triple with a radical of less than 30 but im too lazy to do it.

  • Once one has properly deduced that WLOG 2 divides a, 3 divides b, and 5 divides c, then one merely knows that 30 divides abc, whence rad(abc) is at least 30. Your equal signs are unwarranted. In fact, there's no Pythagorean triple wherefor abc = 30.
I should've said I'm trying to set an absolute minimum for rad(abc).
 
Ahnfeltia

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I wasn't necessarily trying to say that one of them necessarily had 2 as a prime divisor, I'm saying the smallest product of 3 distinct prime numbers is 30 which means that the radical of abc would have to be at least 30. Since a, b, and c are coprime then each would have to have a distinct prime factor. If you try another primitive Pythagorean triple then you would have to change at least one of those prime factors to be a number greater than 2, 3, or 5, which has to have a product greater than 30.
Right. You're right. My bad. I had actually posted a corrected version of the question in which I ask to prove the statement for any Pythagorean triple (not just primitive ones). As you've correctly identified, the claim's pretty uninteresting for primitive Pythagorean triples.
 
trying to ascend

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Given the following polynomial p(x) = x^4 - 3x³ + 2x² - x + 4, with a, b, c and d as its roots, find the value of a^5 + b^5 + c^5 + d^5
 
Ahnfeltia

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Given the following polynomial p(x) = x^4 - 3x³ + 2x² - x + 4, with a, b, c and d as its roots, find the value of a^5 + b^5 + c^5 + d^5
Do you perchance mean a^4 + b^4 + c^4 + d^4?
 
Ahnfeltia

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Given the following polynomial p(x) = x^4 - 3x³ + 2x² - x + 4, with a, b, c and d as its roots, find the value of a^5 + b^5 + c^5 + d^5
We'll find the values of a^n + b^n + c^n + d^n for n = 1 thru 5 in order. But first, some notation. Let e1 = a + b + c + d, e2 = ab + ac + ad + bc + bd + cd, e3 = abc + abd + acd + bcd, and e4 = abcd. By Vieta's formulae, e1 = 3, e2 = 2, e3 = 1, and e4 = 4. Now, a + b + c + d = e1 = 3. Next,
  • a^2 + b^2 + c^2 + d^2 = e1^2 - 2e2 = 3^2 - 2*2 = 5
  • a^3 + b^3 + c^3 + d^3 = e1^3 - 3e1e2 + 3e3 = 3^3 - 3*3*2 + 3*1 = 27 - 18 + 3 = 12
  • a^4 + b^4 + c^4 + d^4 = e1^4 - 4e1^2e2 + 2e2^2 + 4e1e3 - 4e4 = 3^4 - 4*3^2*2 + 2*2^2 + 4*3*1 - 4*4 = 81 - 72 + 8 + 12 - 16 = 13
As for a^5 + b^5 + c^5 + d^5, we know that a^5 = a*a^4 = a(3a^3 - 2a^2 + a - 4) = 3a^4 - 2a^3 + a^2 - 4a as a is zero of p. Similarly for b, c, and d. Ergo, a^5 + b^5 + c^5 + d^5 = 3(a^4 + b^4 + c^4 + d^4) - 2(a^3 + b^3 + c^3 + d^3) + (a^2 + b^2 + c^2 + d^2) - 4(a + b + c + d) = 3*13 - 2*12 + 5 - 3 = 39 - 24 + 2 = 17.

In hindsight using Newton's identities would've been faster...
 
trying to ascend

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We'll find the values of a^n + b^n + c^n + d^n for n = 1 thru 5 in order. But first, some notation. Let e1 = a + b + c + d, e2 = ab + ac + ad + bc + bd + cd, e3 = abc + abd + acd + bcd, and e4 = abcd. By Vieta's formulae, e1 = 3, e2 = 2, e3 = 1, and e4 = 4. Now, a + b + c + d = e1 = 3. Next,
  • a^2 + b^2 + c^2 + d^2 = e1^2 - 2e2 = 3^2 - 2*2 = 5
  • a^3 + b^3 + c^3 + d^3 = e1^3 - 3e1e2 + 3e3 = 3^3 - 3*3*2 + 3*1 = 27 - 18 + 3 = 12
  • a^4 + b^4 + c^4 + d^4 = e1^4 - 4e1^2e2 + 2e2^2 + 4e1e3 - 4e4 = 3^4 - 4*3^2*2 + 2*2^2 + 4*3*1 - 4*4 = 81 - 72 + 8 + 12 - 16 = 13
As for a^5 + b^5 + c^5 + d^5, we know that a^5 = a*a^4 = a(3a^3 - 2a^2 + a - 4) = 3a^4 - 2a^3 + a^2 - 4a as a is zero of p. Similarly for b, c, and d. Ergo, a^5 + b^5 + c^5 + d^5 = 3(a^4 + b^4 + c^4 + d^4) - 2(a^3 + b^3 + c^3 + d^3) + (a^2 + b^2 + c^2 + d^2) - 4(a + b + c + d) = 3*13 - 2*12 + 5 - 3 = 39 - 24 + 2 = 17.

In hindsight using Newton's identities would've been faster...
Correct :feelsokman:

Indeed, the question was meant to be solved using them.
 
Orzmund

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We'll find the values of a^n + b^n + c^n + d^n for n = 1 thru 5 in order. But first, some notation. Let e1 = a + b + c + d, e2 = ab + ac + ad + bc + bd + cd, e3 = abc + abd + acd + bcd, and e4 = abcd. By Vieta's formulae, e1 = 3, e2 = 2, e3 = 1, and e4 = 4. Now, a + b + c + d = e1 = 3. Next,
  • a^2 + b^2 + c^2 + d^2 = e1^2 - 2e2 = 3^2 - 2*2 = 5
  • a^3 + b^3 + c^3 + d^3 = e1^3 - 3e1e2 + 3e3 = 3^3 - 3*3*2 + 3*1 = 27 - 18 + 3 = 12
  • a^4 + b^4 + c^4 + d^4 = e1^4 - 4e1^2e2 + 2e2^2 + 4e1e3 - 4e4 = 3^4 - 4*3^2*2 + 2*2^2 + 4*3*1 - 4*4 = 81 - 72 + 8 + 12 - 16 = 13
As for a^5 + b^5 + c^5 + d^5, we know that a^5 = a*a^4 = a(3a^3 - 2a^2 + a - 4) = 3a^4 - 2a^3 + a^2 - 4a as a is zero of p. Similarly for b, c, and d. Ergo, a^5 + b^5 + c^5 + d^5 = 3(a^4 + b^4 + c^4 + d^4) - 2(a^3 + b^3 + c^3 + d^3) + (a^2 + b^2 + c^2 + d^2) - 4(a + b + c + d) = 3*13 - 2*12 + 5 - 3 = 39 - 24 + 2 = 17.

In hindsight using Newton's identities would've been faster...
How long have you been studying mathematics at such a high level?
 
Ahnfeltia

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In(x)*x -2ln(x)*x + 2x
You forgot a square (the first term should be ln(x)^2*x) and the constant of integration. Probably just a typo/oversight, but still.
 
Ahnfeltia

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How long have you been studying mathematics at such a high level?
That argument wasn't even all that high level. To answer your question, I've been studying mathematical at university level for 6½ years now.
 
Grim_Reaper

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You forgot a square (the first term should be ln(x)^2*x) and the constant of integration. Probably just a typo/oversight, but still.
It was an oversight. I was typing on phone as well.
 
Grim_Reaper

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Fair enough. Typing on phone sucks ass.
Yeah it's because I was in uni and I don't want to log into this site on my laptop with the uni's wifi.
 
Ahnfeltia

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Yeah it's because I was in uni and I don't want to log into this site on my laptop with the uni's wifi.
I doubt your uni would do anything. I've browsed .is on my uni MacBook several times in the past and nothing has happened thus far. I'm an employee btw.
 
Draconian Times

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Mathematics separates bright minds from intellectual impostors. Parrots can appear to be intelligent by paraphrasing something they read in a book, but they could never follow a high level mathematical discussion. That level of abstraction is unbearable for dilettantes.

Although I am well-versed in High School mathematics, I was incapable of pursuing a degree in the field because 1) my passion for the Humanities got the best of me, and 2) I am not quantitatively brilliant, though my verbal IQ is quite high.
 
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Mathematics separates bright minds from intellectual impostors. Parrots can appear to be intelligent by paraphrasing something they read in a book, but they could never follow a high level mathematical discussion. That level of abstraction is unbearable for dilettantes.
I'm an intellectual imposter:feelsrope:.
Although I am well-versed in High School mathematics, I was incapable of pursuing a degree in the field because 1) my passion for the Humanities got the best of me, and 2) I am not quantitatively brilliant, though my verbal IQ is quite high.
High school math and uni math use two completely different skill sets.
 
GriffithIsInnocent

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only when grading
PhD dudes grading my shit used to both be Chinese probably mastered Calculus by 5. nice helpful guys honestly.
 
Draconian Times

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I'm an intellectual imposter:feelsrope:.
I have never claimed to be the smartest person around. Acknowledging one's cognitive limits is a sign of intellectual honesty.

High school math and uni math use two completely different skill sets.
I have come to that realization just by watching lectures on YouTube.
 
GriffithIsInnocent

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I don't think I'm quite as well liked. Too stern.
They weren’t too liked either. Majority of students wanted good grades without putting in effort. Graders said fuck your good grades and have the classes they graded averages of like 45%. Was happy that they were stern because my hard work meant something. Fuck those whiny bitches.
 
Ahnfeltia

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Mathematics separates bright minds from intellectual impostors. Parrots can appear to be intelligent by paraphrasing something they read in a book, but they could never follow a high level mathematical discussion. That level of abstraction is unbearable for dilettantes.
Mathematics is not unique in that regard, as evidenced by the fact that it would fail to categorize you correctly were it unique. Unfortunately most contemporary scientists are of the parrot type you described.
 
Ahnfeltia

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They weren’t too liked either. Majority of students wanted good grades without putting in effort. Graders said fuck your good grades and have the classes they graded averages of like 45%. Was happy that they were stern because my hard work meant something. Fuck those whiny bitches.
How I wish I had more students like you
 
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They weren’t too liked either. Majority of students wanted good grades without putting in effort. Graders said fuck your good grades and have the classes they graded averages of like 45%. Was happy that they were stern because my hard work meant something. Fuck those whiny bitches.
I'm one of those students that don't put any effort but I never complained because it's my fault I received low marks.
 
Orzmund

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They weren’t too liked either. Majority of students wanted good grades without putting in effort. Graders said fuck your good grades and have the classes they graded averages of like 45%. Was happy that they were stern because my hard work meant something. Fuck those whiny bitches.
Parrots might be intuitive but we humans possess an acumen for intuition and abstract concepts.
 
Draconian Times

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Parrots might be intuitive but we humans possess an acumen for intuition and abstract concepts.
True, but we have to concur that most humans are barely above beasts when it comes to rational thought. Bright men we look up to are uncommon. Their innovations have made it far easier for the imbeciles to stay alive.

Ten thousand years ago, the aforementioned imbeciles would have starved to death because they are too incompetent to operate a spear. Now they just go on welfare and the brighter minority of the population carries them through life.
 
GriffithIsInnocent

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Parrots might be intuitive but we humans possess an acumen for intuition and abstract concepts.
Sadly academia consists primarily of parrots. Probably why society is the way it is. Many lack the ability to logically think.
 
GriffithIsInnocent

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I'm one of those students that don't put any effort but I never complained because it's my fault I received low marks.
Based. I can respect that.
 
Orzmund

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True, but we have to concur that most humans are barely above beasts when it comes to rational thought.
Come on, yes they are. Quit being so arrogantly naive. I do not have to concur with your decree as a dog would his master's command.
Bright men we look up to are uncommon. Their innovations have made it far easier for the imbeciles to stay alive.
"Imbeciles," are ordinary people mere idiots?
Ten thousand years ago, the aforementioned imbeciles would have starved to death because they are too incompetent to operate a spear.
That's is not true, if anything the geniuses would die because an excess of intellect ails those who amuse fancies and ideas proliferated by an enhanced affinity to imagine.

You may as well be arguing that an animal is so dimwitted that it would forget its necessary instincts—and the doing so forsake its own will to live.
Now they just go on welfare and the brighter minority of the population carries them through life.
The opposite is true, the genius artists, innovators, and inventors are given the opportunity to exercise their ardour and excess of intellect due to the sacrifices of the masses.

There would be no society of high-brow without an exploited people. I despise the premise that almost everyone else is a absent-minded drone simply because one person is exceptionally talented or intelligent, they have lives too. This adulation of solipism is ridiculous.


One last inquiry: do you think you are the exception? the savant who the world exists to experience?

This entire thread consists of wanton sophistry and conceited affirmations. It is actually irritating me. These people are smarter than parrots.
 

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