May 14th, 2018, 10:13 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 394 Thanks: 27 Math Focus: Number theory  Equations solvable only by directly providing the answer
Are there equations with solutions, but totally unsolvable mathematically unless by substituting simple variables with given complex answers?

May 14th, 2018, 11:22 PM  #2  
Senior Member Joined: Aug 2012 Posts: 2,101 Thanks: 605  Quote:
One topic that comes to mind is the solution of polynomials. We know that there is no general algebraic procedure to solve polynomials of fifth degree and higher; but every such polynomial has zeros by the fundamental theorem of algebra. In other words we can prove there are solutions but there is no procedural way to find them. The best we can do is iterative approximation. Is that what you mean?  
May 15th, 2018, 12:00 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 19,982 Thanks: 1853 
How about the equation x² = 1?

May 15th, 2018, 10:15 AM  #4  
Senior Member Joined: May 2015 From: Arlington, VA Posts: 394 Thanks: 27 Math Focus: Number theory  Quote:
Do algebraic functions include the methods for finding most zeros, next to transcendental and "beyond"? Or do exponential and trigonometric functions include just as many possible zeros? Otherwise, would all of these three groups of functions (sans transcendental) follow the fundamental theorem of algebra?  
May 15th, 2018, 11:24 AM  #5  
Senior Member Joined: Aug 2012 Posts: 2,101 Thanks: 605  Quote:
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FTA only applies to polynomials. Last edited by skipjack; July 21st, 2018 at 10:08 PM.  
May 15th, 2018, 03:18 PM  #6 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 394 Thanks: 27 Math Focus: Number theory 
Maschke, Do algebraic functions include the methods for finding most zeros, next to transcendental and "beyond"? >Is the set greater for the real numbers than that for transcendental numbers plus algebraic numbers? Or do exponential and trigonometric functions include just as many possible zeros? >Are polynomial, exponential and trigonometric functions of exclusive yet equivalent sets  and with what cardinality? Many mathematicians died before age 40. What's the omen of hearse #40? 
May 16th, 2018, 08:09 AM  #7 
Senior Member Joined: May 2008 Posts: 297 Thanks: 81  
May 16th, 2018, 12:38 PM  #8 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,624 Thanks: 954  
May 25th, 2018, 01:59 PM  #9  
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894  Quote:
be expressed in other ways. Quote:
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May 25th, 2018, 04:12 PM  #10  
Senior Member Joined: Aug 2012 Posts: 2,101 Thanks: 605  Quote:
Unless they do pioneering work on the quintic. That's generally fatal.  

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