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Although the bookmark function can save my favorite questions, it is still a mess since I have bookmarked thousands of questions for a long time.

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Yes, there's an advanced search option for this: just add inbookmarks:mine to your search query.

You can even search within another user's bookmarks; I don't have any bookmarks on the main site, so when I search for inbookmarks:mine integration I get no results, but this must be the results you get:

enter image description here

The ability to group them has been requested (a couple of times): Allow grouping of favorite questions but it doesn't seem the company wants to spend time on this. You can combine bookmark search with tags, and if you want to categorize them in a different way, there's always the option to use your browser's bookmark function.

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I keep two text files at home, here are selections. I group them by what I see as the topic. ANYONE can do this. Sometimes an OP wants lots of examples in a topic I have frequently answered. I just copy a relevant section and paste into an answer window, say it is too long for a comment and make it CW.

Vieta Jumping/ Hurwitz/Markov         Grundlösung
 http://math.stackexchange.com/questions/115272/let-x-and-y-be-positive-integers-such-that-xy-mid-x2y21?rq=1
http://math.stackexchange.com/questions/632097/diophantine-quartic-equation-in-four-variables/632178#632178
http://math.stackexchange.com/questions/829228/is-it-true-that-fx-y-dfracx2y2xy-t-has-only-finitely-many-distinct-i/1480206#1480206  !!!!!!!!!  +++++++  
http://math.stackexchange.com/questions/1411049/if-a-b-are-positive-integers-and-ab-1-mid-a%C2%B2-b%C2%B2-then-prove-that-q  
http://math.stackexchange.com/questions/1415252/proving-that-all-terms-in-sequence-are-positive-integers-recursive/1415304#1415304
http://math.stackexchange.com/questions/1417404/diophantine-equation-xyxy1-kxy-0
http://math.stackexchange.com/questions/1462435/find-the-postive-integers-such-xyxy-mid-x2y2-2
http://math.stackexchange.com/questions/1478289/find-the-integer-values-of-c
http://math.stackexchange.com/questions/1604292/find-all-solutions-to-the-diophantine-equation-x2y2z2-xyz22/1604582#1604582
http://math.stackexchange.com/questions/1780420/characterize-the-integers-a-b-satisfying-ab-1a2b2/1781381#1781381
http://math.stackexchange.com/questions/1930438/equation-with-vieta-jumping-xyz2-nxyz/1930796#1930796     Follows Hurwitz very closely!!  (x+y+z)^2 = nxyz   
http://math.stackexchange.com/questions/2188567/showing-that-m2-n21-is-a-square/2193043#2193043  LEMMA + 
http://math.stackexchange.com/questions/2188567/showing-that-m2-n21-is-a-square/2193944#2193944   LEMMA -
https://math.stackexchange.com/questions/2399830/how-prove-infinitely-many-postive-integers-triples-x-y-z-such-xyz22x/2400509#2400509  
https://math.stackexchange.com/questions/2664145/find-all-possible-value-of-c/2668331#2668331 
https://math.stackexchange.com/questions/2818849/what-are-the-solutions-of-the-equation-3np3n2-n2p2-with-n-and-p-positive/2818883#2818883
https://math.stackexchange.com/questions/2853669/prove-there-is-no-x-y-in-mathbb-z-text-satisfying-fracxy-fracy/2853742#2853742
https://math.stackexchange.com/questions/2869251/integer-points-on-a-hyperbola/2869365#2869365 two spiral arms 
https://math.stackexchange.com/questions/2955986/mnp-1-2-sqrtmnp-in-positive-integers
https://artofproblemsolving.com/community/c6h1726220_if_fraction_integer_then_equal_to_5   Poland 1991 training


     Pell
    http://math.stackexchange.com/questions/1864263/find-general-solution-for-the-equation-1-2-cdots-n-%E2%88%92-1-n-1-n/1864611#1864611
    https://math.stackexchange.com/questions/2424587/when-is-5n214n1-a-perfect-square/2424599#2424599  x^2 - 5 y^2 =44
    https://math.stackexchange.com/questions/2546839/state-2-values-of-x-for-which-the-value-of-3x24x-14-is-a-perfect-square/2546875#2546875
       x^2 - 3 y^2 = 46
    
    TOPOGRAPH 
    RIVERBEND PROOF:
    http://math.stackexchange.com/questions/342284/generate-solutions-of-quadratic-diophantine-equation/346821#346821
    diagrams
    http://math.stackexchange.com/questions/81917/another-quadratic-diophantine-equation-how-do-i-proceed/144794#144794
    http://math.stackexchange.com/questions/228356/how-to-find-solutions-of-x2-3y2-2/228405#228405
    http://math.stackexchange.com/questions/342284/generate-solutions-of-quadratic-diophantine-equation/345128#345128
    http://math.stackexchange.com/questions/487051/why-cant-the-alpertron-solve-this-pell-like-equation/487063#487063
    http://math.stackexchange.com/questions/512621/finding-all-solutions-of-the-pell-type-equation-x2-5y2-4/512649#512649
    http://math.stackexchange.com/questions/680972/if-m-n-in-mathbb-z-2-satisfies-3m2m-4n2n-then-m-n-is-a-perfect-square/686351#686351
    http://math.stackexchange.com/questions/739752/how-to-solve-binary-form-ax2bxycy2-m-for-integer-and-rational-x-y/739765#739765  ::::  69   55
    http://math.stackexchange.com/questions/742181/find-all-integer-solutions-for-the-equation-5x2-y2-4/756972#756972
    http://math.stackexchange.com/questions/822503/positive-integer-n-such-that-2n1-3n1-are-both-perfect-squares/822517#822517
    http://math.stackexchange.com/questions/1078450/maps-of-primitive-vectors-and-conways-river-has-anyone-built-this-in-sage/1078979#1078979
    http://math.stackexchange.com/questions/1091310/infinitely-many-systems-of-23-consecutive-integers/1093382#1093382
    http://math.stackexchange.com/questions/1132187/solve-the-following-equation-for-x-and-y/1132347#1132347  <1,-1,-1>
    http://math.stackexchange.com/questions/1132799/finding-integers-of-the-form-3x2-xy-5y2-where-x-and-y-are-integers
    http://math.stackexchange.com/questions/1221178/small-integral-representation-as-x2-2y2-in-pells-equation/1221280#1221280
    http://math.stackexchange.com/questions/1404023/solving-the-equation-x2-7y2-3-over-integers/1404126#1404126
    http://math.stackexchange.com/questions/1599211/solutions-to-diophantine-equations/1600010#1600010
    http://math.stackexchange.com/questions/1667323/how-to-prove-that-the-roots-of-this-equation-are-integers/1667380#1667380
    http://math.stackexchange.com/questions/1719280/does-the-pell-like-equation-x2-dy2-k-have-a-simple-recursion-like-x2-dy2
    http://math.stackexchange.com/questions/1737385/if-d1-is-a-squarefree-integer-show-that-x2-dy2-c-gives-some-bounds-i/1737824#1737824 "seeds"
    http://math.stackexchange.com/questions/1772594/find-all-natural-numbers-n-such-that-21n2-20-is-a-perfect-square/1773319#1773319
    https://math.stackexchange.com/questions/2549380/is-there-a-simple-proof-that-if-b-aba-ab-1-then-a-b-must-be-fibon/2549440#2549440 1,1,-1; 1,11
    https://math.stackexchange.com/questions/2579293/to-find-all-integral-solutions-of-3x2-4y2-11/2579305#2579305
    https://math.stackexchange.com/questions/3778825/how-do-we-solve-pell-like-equations/3778953#3778953
    https://math.stackexchange.com/questions/3804117/diophantine-equation-x2-xy-%e2%88%92-3y2-17/3804236#3804236 <1,1,-3>
    
    http://www.maths.ed.ac.uk/~aar/papers/conwaysens.pdf   (Conway) 
    https://www.math.cornell.edu/~hatcher/TN/TNbook.pdf    (Hatcher)
    http://bookstore.ams.org/mbk-105/                  (Weissman)
    http://www.springer.com/us/book/9780387955872             (Stillwell)
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