This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/Debian) (preloaded format=pdflatex 2015.12.28)  14 DEC 2016 00:12
entering extended mode
 restricted \write18 enabled.
 %&-line parsing enabled.
**problem-4.tex
(./problem-4.tex
LaTeX2e <2015/01/01>
Babel <3.9l> and hyphenation patterns for 79 languages loaded.
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2014/09/29 v1.4h Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo
File: size10.clo 2014/09/29 v1.4h Standard LaTeX file (size option)
)
\c@part=\count79
\c@section=\count80
\c@subsection=\count81
\c@subsubsection=\count82
\c@paragraph=\count83
\c@subparagraph=\count84
\c@figure=\count85
\c@table=\count86
\abovecaptionskip=\skip41
\belowcaptionskip=\skip42
\bibindent=\dimen102
)
\c@theorem=\count87
 (./problem-4.aux)
\openout1 = `problem-4.aux'.

LaTeX Font Info:    Checking defaults for OML/cmm/m/it on input line 7.
LaTeX Font Info:    ... okay on input line 7.
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LaTeX Font Info:    ... okay on input line 7.
LaTeX Font Info:    Checking defaults for OMS/cmsy/m/n on input line 7.
LaTeX Font Info:    ... okay on input line 7.
LaTeX Font Info:    Checking defaults for OMX/cmex/m/n on input line 7.
LaTeX Font Info:    ... okay on input line 7.
LaTeX Font Info:    Checking defaults for U/cmr/m/n on input line 7.
LaTeX Font Info:    ... okay on input line 7.
LaTeX Font Info:    External font `cmex10' loaded for size
(Font)              <12> on input line 8.
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LaTeX Font Info:    External font `cmex10' loaded for size
(Font)              <5> on input line 13.

! Missing $ inserted.
<inserted text> 
                $
l.15 \begin{itemize}
                    
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Missing \endgroup inserted.
<inserted text> 
                \endgroup 
l.15 \begin{itemize}
                    
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.


! LaTeX Error: Lonely \item--perhaps a missing list environment.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.16 \item $
            1^{10^{10^{10}}} + 4 = 5$,
Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.


! LaTeX Error: Lonely \item--perhaps a missing list environment.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.17 \item $
            3 + 2 = 5$,
Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.


! LaTeX Error: Lonely \item--perhaps a missing list environment.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.18 \item $
            2^2 + 1 = 5$, and shockingly,
Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.


! LaTeX Error: Lonely \item--perhaps a missing list environment.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.19 \item $
            2 + 3^{10}\cdot 109  = 23^5$.
Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Runaway argument?
{itemize 
! Paragraph ended before \end was complete.
<to be read again> 
                   \par 
l.21 
     
I suspect you've forgotten a `}', causing me to apply this
control sequence to too much text. How can we recover?
My plan is to forget the whole thing and hope for the best.

Runaway argument?
Klutz! If $E$ is an elliptic curve over $\matbf {Q}$, then $$ {\rm or\ETC.
! Paragraph ended before \@ythm was complete.
<to be read again> 
                   \par 
l.31 
     
I suspect you've forgotten a `}', causing me to apply this
control sequence to too much text. How can we recover?
My plan is to forget the whole thing and hope for the best.


! LaTeX Error: \begin{theorem} on input line 33 ended by \end{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.43 \end{document}
                   
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

[1

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(\end occurred inside a group at level 2)

### semi simple group (level 2) entered at line 33 (\begingroup)
### semi simple group (level 1) entered at line 25 (\begingroup)
### bottom level 
Here is how much of TeX's memory you used:
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Output written on problem-4.pdf (1 page, 127236 bytes).
PDF statistics:
 56 PDF objects out of 1000 (max. 8388607)
 40 compressed objects within 1 object stream
 0 named destinations out of 1000 (max. 500000)
 1 words of extra memory for PDF output out of 10000 (max. 10000000)