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Is there a simple example that empirical evidence is misleading?

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3

$begingroup$

Suppose that I want to show a student that emperical evidence in mathematics is not enough and we do need proofs, what kind of examples can I use?

By emperical evidence, I mean (most of the time) you cannot simply check the statement $S(n)$ for $n in 1,dots, 10^9$ and conclude it's true for all $n in mathbb N$.

undergraduate-education

share|improve this question

edited 3 hours ago

Rusty Core

18319

asked 9 hours ago

ablmfablmf

24219

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  math.stackexchange.com/questions/514/…
  $endgroup$
  – Jasper
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  mathoverflow.net/q/15444/36173
  $endgroup$
  – Paracosmiste
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If you pick $n$ (generic) points on a circle and connect them with lines, the disc divides into a number of regions. It appears to be a power of two for $nleq 5$, then it changes. See the nice article: quantamagazine.org/…
  $endgroup$
  – Adam
  2 hours ago
  

  
  
  
  

add a comment | 

3

$begingroup$

Suppose that I want to show a student that emperical evidence in mathematics is not enough and we do need proofs, what kind of examples can I use?

By emperical evidence, I mean (most of the time) you cannot simply check the statement $S(n)$ for $n in 1,dots, 10^9$ and conclude it's true for all $n in mathbb N$.

undergraduate-education

share|improve this question

edited 3 hours ago

Rusty Core

18319

asked 9 hours ago

ablmfablmf

24219

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  math.stackexchange.com/questions/514/…
  $endgroup$
  – Jasper
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  mathoverflow.net/q/15444/36173
  $endgroup$
  – Paracosmiste
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If you pick $n$ (generic) points on a circle and connect them with lines, the disc divides into a number of regions. It appears to be a power of two for $nleq 5$, then it changes. See the nice article: quantamagazine.org/…
  $endgroup$
  – Adam
  2 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

Suppose that I want to show a student that emperical evidence in mathematics is not enough and we do need proofs, what kind of examples can I use?

By emperical evidence, I mean (most of the time) you cannot simply check the statement $S(n)$ for $n in 1,dots, 10^9$ and conclude it's true for all $n in mathbb N$.

undergraduate-education

share|improve this question

edited 3 hours ago

Rusty Core

18319

asked 9 hours ago

ablmfablmf

24219

$endgroup$

share|improve this question

edited 3 hours ago

Rusty Core

18319

asked 9 hours ago

ablmfablmf

24219

share|improve this question

edited 3 hours ago

Rusty Core

18319

asked 9 hours ago

ablmfablmf

24219

edited 3 hours ago

Rusty Core

18319

edited 3 hours ago

Rusty Core

18319

asked 9 hours ago

ablmfablmf

24219

asked 9 hours ago

ablmfablmf

24219

2 Answers
2

active

oldest

votes

3

$begingroup$

There are some collections of such examples at sister sites:

• Conjectures that have been disproved with extremely large counterexamples?
  at Mathematics Stack Exchange.




• Examples of eventual counterexamples at MathOverflow.

---

One rather simple example that can be checked with a calculator is the conjecture by Fermat, that all numbers of the form $$2^2^n+1, qquad n in mathbb N_0$$ are prime.

In fact,

• 
  $2^2^0 +1 = 3$ is prime


• 
  $2^2^1 +1 = 5$ is prime


• 
  $2^2^2 +1 = 17$ is prime


• 
  $2^2^3 +1 = 257$ is prime


• 
  $2^2^4 +1 = 65537$ is prime


• 
  $2^2^5 +1 = 4294967297$ is not prime: $4294967297 = 641 cdot 6700417$
  

So the original conjecture is clearly false, but it took nearly 100 years to find the counterexample. All following Fermat numbers appear to be composite, but this is an open problem.

share|improve this answer

answered 5 hours ago

JasperJasper

799513

$endgroup$

add a comment | 

3

$begingroup$

Strangely, just this morning I asked Wolfram|Alpha to compute the sum $$sum_n=1^inftyfrac1nsin(n)$$ and it returned the approximate value of $-0.863507$. I asked it to "show more digits", and it returned a new approximation:

$94.377284731050845020943145217217734512865979242824685504875914407196948018$

I was trying to illustrate a series whose convergence (or divergence) is difficult to determine and was treated to some very different approximations. Note that Wolfram did not tell whether the series converges.

share|improve this answer

edited 4 hours ago

answered 8 hours ago

Nick CNick C

2,260626

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  FWIW: You can see in MSE 665776 that the sequence $frac1n sin(n)$ does not converge; in particular, the sequence does not converge to $0$. So, the series described here does not converge.
  $endgroup$
  – Benjamin Dickman
  26 mins ago
  

  
  
  
  

add a comment | 

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3

$begingroup$

There are some collections of such examples at sister sites:

• Conjectures that have been disproved with extremely large counterexamples?
  at Mathematics Stack Exchange.




• Examples of eventual counterexamples at MathOverflow.

---

One rather simple example that can be checked with a calculator is the conjecture by Fermat, that all numbers of the form $$2^2^n+1, qquad n in mathbb N_0$$ are prime.

In fact,

• 
  $2^2^0 +1 = 3$ is prime


• 
  $2^2^1 +1 = 5$ is prime


• 
  $2^2^2 +1 = 17$ is prime


• 
  $2^2^3 +1 = 257$ is prime


• 
  $2^2^4 +1 = 65537$ is prime


• 
  $2^2^5 +1 = 4294967297$ is not prime: $4294967297 = 641 cdot 6700417$
  

So the original conjecture is clearly false, but it took nearly 100 years to find the counterexample. All following Fermat numbers appear to be composite, but this is an open problem.

share|improve this answer

answered 5 hours ago

JasperJasper

799513

$endgroup$

add a comment | 

3

$begingroup$

There are some collections of such examples at sister sites:

• Conjectures that have been disproved with extremely large counterexamples?
  at Mathematics Stack Exchange.




• Examples of eventual counterexamples at MathOverflow.

---

One rather simple example that can be checked with a calculator is the conjecture by Fermat, that all numbers of the form $$2^2^n+1, qquad n in mathbb N_0$$ are prime.

In fact,

• 
  $2^2^0 +1 = 3$ is prime


• 
  $2^2^1 +1 = 5$ is prime


• 
  $2^2^2 +1 = 17$ is prime


• 
  $2^2^3 +1 = 257$ is prime


• 
  $2^2^4 +1 = 65537$ is prime


• 
  $2^2^5 +1 = 4294967297$ is not prime: $4294967297 = 641 cdot 6700417$
  

So the original conjecture is clearly false, but it took nearly 100 years to find the counterexample. All following Fermat numbers appear to be composite, but this is an open problem.

share|improve this answer

answered 5 hours ago

JasperJasper

799513

$endgroup$

add a comment | 

$begingroup$

There are some collections of such examples at sister sites:

• Conjectures that have been disproved with extremely large counterexamples?
  at Mathematics Stack Exchange.




• Examples of eventual counterexamples at MathOverflow.

---

One rather simple example that can be checked with a calculator is the conjecture by Fermat, that all numbers of the form $$2^2^n+1, qquad n in mathbb N_0$$ are prime.

In fact,

• 
  $2^2^0 +1 = 3$ is prime


• 
  $2^2^1 +1 = 5$ is prime


• 
  $2^2^2 +1 = 17$ is prime


• 
  $2^2^3 +1 = 257$ is prime


• 
  $2^2^4 +1 = 65537$ is prime


• 
  $2^2^5 +1 = 4294967297$ is not prime: $4294967297 = 641 cdot 6700417$
  

So the original conjecture is clearly false, but it took nearly 100 years to find the counterexample. All following Fermat numbers appear to be composite, but this is an open problem.

share|improve this answer

answered 5 hours ago

JasperJasper

799513

$endgroup$

share|improve this answer

answered 5 hours ago

JasperJasper

799513

answered 5 hours ago

JasperJasper

799513

answered 5 hours ago

JasperJasper

799513

3

$begingroup$

Strangely, just this morning I asked Wolfram|Alpha to compute the sum $$sum_n=1^inftyfrac1nsin(n)$$ and it returned the approximate value of $-0.863507$. I asked it to "show more digits", and it returned a new approximation:

$94.377284731050845020943145217217734512865979242824685504875914407196948018$

I was trying to illustrate a series whose convergence (or divergence) is difficult to determine and was treated to some very different approximations. Note that Wolfram did not tell whether the series converges.

share|improve this answer

edited 4 hours ago

answered 8 hours ago

Nick CNick C

2,260626

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  FWIW: You can see in MSE 665776 that the sequence $frac1n sin(n)$ does not converge; in particular, the sequence does not converge to $0$. So, the series described here does not converge.
  $endgroup$
  – Benjamin Dickman
  26 mins ago
  

  
  
  
  

add a comment | 

3

$begingroup$

Strangely, just this morning I asked Wolfram|Alpha to compute the sum $$sum_n=1^inftyfrac1nsin(n)$$ and it returned the approximate value of $-0.863507$. I asked it to "show more digits", and it returned a new approximation:

$94.377284731050845020943145217217734512865979242824685504875914407196948018$

I was trying to illustrate a series whose convergence (or divergence) is difficult to determine and was treated to some very different approximations. Note that Wolfram did not tell whether the series converges.

share|improve this answer

edited 4 hours ago

answered 8 hours ago

Nick CNick C

2,260626

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  FWIW: You can see in MSE 665776 that the sequence $frac1n sin(n)$ does not converge; in particular, the sequence does not converge to $0$. So, the series described here does not converge.
  $endgroup$
  – Benjamin Dickman
  26 mins ago
  

  
  
  
  

add a comment | 

$begingroup$

Strangely, just this morning I asked Wolfram|Alpha to compute the sum $$sum_n=1^inftyfrac1nsin(n)$$ and it returned the approximate value of $-0.863507$. I asked it to "show more digits", and it returned a new approximation:

$94.377284731050845020943145217217734512865979242824685504875914407196948018$

I was trying to illustrate a series whose convergence (or divergence) is difficult to determine and was treated to some very different approximations. Note that Wolfram did not tell whether the series converges.

share|improve this answer

edited 4 hours ago

answered 8 hours ago

Nick CNick C

2,260626

$endgroup$

share|improve this answer

edited 4 hours ago

answered 8 hours ago

Nick CNick C

2,260626

answered 8 hours ago

Nick CNick C

2,260626

answered 8 hours ago

Nick CNick C

2,260626