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Type | Label | Description |
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Statement | ||
Theorem | eldioph4i 35701* | Forward-only version of eldioph4b 35700. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
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Theorem | diophren 35702* | Change variables in a Diophantine set, using class notation. This allows already proved Diophantine sets to be reused in contexts with more variables. (Contributed by Stefan O'Rear, 16-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.) |
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Theorem | rabrenfdioph 35703* | Change variable numbers in a Diophantine class abstraction using explicit substitution. (Contributed by Stefan O'Rear, 17-Oct-2014.) |
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Theorem | rabren3dioph 35704* | Change variable numbers in a 3-variable Diophantine class abstraction. (Contributed by Stefan O'Rear, 17-Oct-2014.) |
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Theorem | fphpd 35705* | Pigeonhole principle expressed with implicit substitution. If the range is smaller than the domain, two inputs must be mapped to the same output. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
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Theorem | fphpdo 35706* | Pigeonhole principle for sets of real numbers with implicit output reordering. (Contributed by Stefan O'Rear, 12-Sep-2014.) |
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Theorem | ctbnfien 35707 | An infinite subset of a countable set is countable, without using choice. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
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Theorem | fiphp3d 35708* | Infinite pigeonhole principle for partitioning an infinite set between finitely many buckets. (Contributed by Stefan O'Rear, 18-Oct-2014.) |
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Theorem | rencldnfilem 35709* | Lemma for rencldnfi 35710. (Contributed by Stefan O'Rear, 18-Oct-2014.) |
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Theorem | rencldnfi 35710* | A set of real numbers which comes arbitrarily close to some target yet excludes it is infinite. The work is done in rencldnfilem 35709 using infima; this theorem removes the requirement that A be nonempty. (Contributed by Stefan O'Rear, 19-Oct-2014.) |
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Theorem | modelico 35711 | Modular reduction produces a half-open interval. (Contributed by Stefan O'Rear, 12-Sep-2014.) |
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Theorem | irrapxlem1 35712* |
Lemma for irrapx1 35718. Divides the unit interval into ![]() ![]() |
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Theorem | irrapxlem2 35713* | Lemma for irrapx1 35718. Two multiples in the same bucket means they are very close mod 1. (Contributed by Stefan O'Rear, 12-Sep-2014.) |
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Theorem | irrapxlem3 35714* | Lemma for irrapx1 35718. By subtraction, there is a multiple very close to an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
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Theorem | irrapxlem4 35715* |
Lemma for irrapx1 35718. Eliminate ranges, use positivity of the
input to
force positivity of the output by increasing ![]() |
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Theorem | irrapxlem5 35716* | Lemma for irrapx1 35718. Switching to real intervals and fraction syntax. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
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Theorem | irrapxlem6 35717* | Lemma for irrapx1 35718. Explicit description of a non-closed set. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
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Theorem | irrapx1 35718* | Dirichlet's approximation theorem. Every positive irrational number has infinitely many rational approximations which are closer than the inverse squares of their reduced denominators. Lemma 61 in [vandenDries] p. 42. (Contributed by Stefan O'Rear, 14-Sep-2014.) |
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Theorem | pellexlem1 35719 | Lemma for pellex 35725. Arithmetical core of pellexlem3, norm lower bound. This begins Dirichlet's proof of the Pell equation solution existence; the proof here follows theorem 62 of [vandenDries] p. 43. (Contributed by Stefan O'Rear, 14-Sep-2014.) |
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Theorem | pellexlem2 35720 | Lemma for pellex 35725. Arithmetical core of pellexlem3, norm upper bound. (Contributed by Stefan O'Rear, 14-Sep-2014.) |
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Theorem | pellexlem3 35721* |
Lemma for pellex 35725. To each good rational approximation of
![]() ![]() ![]() ![]() ![]() |
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Theorem | pellexlem4 35722* | Lemma for pellex 35725. Invoking irrapx1 35718, we have infinitely many near-solutions. (Contributed by Stefan O'Rear, 14-Sep-2014.) |
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Theorem | pellexlem5 35723* | Lemma for pellex 35725. Invoking fiphp3d 35708, we have infinitely many near-solutions for some specific norm. (Contributed by Stefan O'Rear, 19-Oct-2014.) |
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Theorem | pellexlem6 35724* | Lemma for pellex 35725. Doing a field division between near solutions get us to norm 1, and the modularity constraint ensures we still have an integer. Returning NN guarantees that we are not returning the trivial solution (1,0). We are not explicitly defining the Pell-field, Pell-ring, and Pell-norm explicitly because after this construction is done we will never use them. This is mostly basic algebraic number theory and could be simplified if a generic framework for that were in place. (Contributed by Stefan O'Rear, 19-Oct-2014.) |
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Theorem | pellex 35725* | Every Pell equation has a nontrivial solution. Theorem 62 in [vandenDries] p. 43. (Contributed by Stefan O'Rear, 19-Oct-2014.) |
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Syntax | csquarenn 35726 | Extend class notation to include the set of square positive integers. |
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Syntax | cpell1qr 35727 | Extend class notation to include the class of quadrant-1 Pell solutions. |
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Syntax | cpell1234qr 35728 | Extend class notation to include the class of any-quadrant Pell solutions. |
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Syntax | cpell14qr 35729 | Extend class notation to include the class of positive Pell solutions. |
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Syntax | cpellfund 35730 | Extend class notation to include the Pell-equation fundamental solution function. |
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Syntax | cpellfundold 35731 | Extend class notation to include the Pell-equation fundamental solution function (old version). |
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Definition | df-squarenn 35732 | Define the set of square positive integers. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Definition | df-pell1qr 35733* | Define the solutions of a Pell equation in the first quadrant. To avoid pair pain, we represent this via the canonical embedding into the reals. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
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Definition | df-pell14qr 35734* | Define the positive solutions of a Pell equation. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
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Definition | df-pell1234qr 35735* | Define the general solutions of a Pell equation. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
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Definition | df-pellfund 35736* | A function mapping Pell discriminants to the corresponding fundamental solution. (Contributed by Stefan O'Rear, 18-Sep-2014.) (Revised by AV, 17-Sep-2020.) |
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Definition | df-pellfundOLD 35737* | A function mapping Pell discriminants to the corresponding fundamental solution. (Contributed by Stefan O'Rear, 18-Sep-2014.) Obsolete version of df-pellfund 35736 as of 17-Sep-2020. (New usage is discouraged.) |
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Theorem | pell1qrval 35738* | Value of the set of first-quadrant Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
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Theorem | elpell1qr 35739* | Membership in a first-quadrant Pell solution set. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
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Theorem | pell14qrval 35740* | Value of the set of positive Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
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Theorem | elpell14qr 35741* | Membership in the set of positive Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
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Theorem | pell1234qrval 35742* | Value of the set of general Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
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Theorem | elpell1234qr 35743* | Membership in the set of general Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
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Theorem | pell1234qrre 35744 | General Pell solutions are (coded as) real numbers. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
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Theorem | pell1234qrne0 35745 | No solution to a Pell equation is zero. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
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Theorem | pell1234qrreccl 35746 | General solutions of the Pell equation are closed under reciprocals. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Theorem | pell1234qrmulcl 35747 | General solutions of the Pell equation are closed under multiplication. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Theorem | pell14qrss1234 35748 | A positive Pell solution is a general Pell solution. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Theorem | pell14qrre 35749 | A positive Pell solution is a real number. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Theorem | pell14qrne0 35750 | A positive Pell solution is a nonzero number. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
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Theorem | pell14qrgt0 35751 | A positive Pell solution is a positive number. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Theorem | pell14qrrp 35752 | A positive Pell solution is a positive real. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
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Theorem | pell1234qrdich 35753 | A general Pell solution is either a positive solution, or its negation is. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Theorem | elpell14qr2 35754 | A number is a positive Pell solution iff it is positive and a Pell solution, justifying our name choice. (Contributed by Stefan O'Rear, 19-Oct-2014.) |
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Theorem | pell14qrmulcl 35755 | Positive Pell solutions are closed under multiplication. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
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Theorem | pell14qrreccl 35756 | Positive Pell solutions are closed under reciprocal. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Theorem | pell14qrdivcl 35757 | Positive Pell solutions are closed under division. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Theorem | pell14qrexpclnn0 35758 | Lemma for pell14qrexpcl 35759. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Theorem | pell14qrexpcl 35759 | Positive Pell solutions are closed under integer powers. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Theorem | pell1qrss14 35760 | First-quadrant Pell solutions are a subset of the positive solutions. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Theorem | pell14qrdich 35761 | A positive Pell solution is either in the first quadrant, or its reciprocal is. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Theorem | pell1qrge1 35762 | A Pell solution in the first quadrant is at least 1. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
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Theorem | pell1qr1 35763 | 1 is a Pell solution and in the first quadrant as one. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
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Theorem | elpell1qr2 35764 | The first quadrant solutions are precisely the positive Pell solutions which are at least one. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Theorem | pell1qrgaplem 35765 | Lemma for pell1qrgap 35766. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Theorem | pell1qrgap 35766 | First-quadrant Pell solutions are bounded away from 1. (This particular bound allows us to prove exact values for the fundamental solution later.) (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Theorem | pell14qrgap 35767 | Positive Pell solutions are bounded away from 1. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Theorem | pell14qrgapw 35768 | Positive Pell solutions are bounded away from 1, with a friendlier bound. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Theorem | pellqrexplicit 35769 | Condition for a calculated real to be a Pell solution. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
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Theorem | infmrgelbi 35770* | Any lower bound of a nonempty set of real numbers is less than or equal to its infimum, one-direction version. (Contributed by Stefan O'Rear, 1-Sep-2013.) (Revised by AV, 17-Sep-2020.) |
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Theorem | infmrgelbiOLD 35771* | Any lower bound of a nonempty set of real numbers is less than or equal to its infimum, one-direction version. (Contributed by Stefan O'Rear, 1-Sep-2013.) Obsolete version of infmrgelbi 35770 as of 17-Sep-2020. ( (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | pellqrex 35772* | There is a nontrivial solution of a Pell equation in the first quadrant. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Theorem | pellfundval 35773* | Value of the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 18-Sep-2014.) (Revised by AV, 17-Sep-2020.) |
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Theorem | pellfundvalOLD 35774* | Value of the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 18-Sep-2014.) Obsolete version of pellfundval 35773 as of 17-Sep-2020. ( (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | pellfundre 35775 | The fundamental solution of a Pell equation exists as a real number. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Theorem | pellfundge 35776 | Lower bound on the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
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Theorem | pellfundgt1 35777 | Weak lower bound on the Pell fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
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Theorem | pellfundlb 35778 | A nontrivial first quadrant solution is at least as large as the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Proof shortened by AV, 15-Sep-2020.) |
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Theorem | pellfundglb 35779* | If a real is larger than the fundamental solution, there is a nontrivial solution less than it. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Theorem | pellfundex 35780 |
The fundamental solution as an infimum is itself a solution, showing
that the solution set is discrete.
Since the fundamental solution is an infimum, there must be an element ge to Fund and lt 2*Fund. If this element is equal to the fundamental solution we're done, otherwise use the infimum again to find another element which must be ge Fund and lt the first element; their ratio is a group element in (1,2), contradicting pell14qrgapw 35768. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Theorem | pellfund14gap 35781 | There are no solutions between 1 and the fundamental solution. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
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Theorem | pellfundrp 35782 | The fundamental Pell solution is a positive real. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
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Theorem | pellfundne1 35783 | The fundamental Pell solution is never 1. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
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Section should be obsolete because its contents are covered by section "Logarithms to an arbitrary base" now. | ||
Theorem | reglogcl 35784 | General logarithm is a real number. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbcl 23766 instead. |
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Theorem | reglogltb 35785 | General logarithm preserves "less than". (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logblt 23777 instead. |
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Theorem | reglogleb 35786 |
General logarithm preserves ![]() |
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Theorem | reglogmul 35787 | Multiplication law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbmul 23770 instead. |
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Theorem | reglogexp 35788 | Power law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbzexp 23769 instead. |
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Theorem | reglogbas 35789 | General log of the base is 1. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logbid1 23761 instead. |
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Theorem | reglog1 35790 | General log of 1 is 0. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logb1 23762 instead. |
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Theorem | reglogexpbas 35791 | General log of a power of the base is the exponent. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbexp 23773 instead. |
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Theorem | pellfund14 35792* | Every positive Pell solution is a power of the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
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Theorem | pellfund14b 35793* | The positive Pell solutions are precisely the integer powers of the fundamental solution. To get the general solution set (which we will not be using), throw in a copy of Z/2Z. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
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Syntax | crmx 35794 | Extend class notation to include the Robertson-Matiyasevich X sequence. |
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Syntax | crmy 35795 | Extend class notation to include the Robertson-Matiyasevich Y sequence. |
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Definition | df-rmx 35796* | Define the X sequence as the rational part of some solution of a special Pell equation. See frmx 35807 and rmxyval 35809 for a more useful but non-eliminable definition. (Contributed by Stefan O'Rear, 21-Sep-2014.) |
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Definition | df-rmy 35797* | Define the X sequence as the irrational part of some solution of a special Pell equation. See frmy 35808 and rmxyval 35809 for a more useful but non-eliminable definition. (Contributed by Stefan O'Rear, 21-Sep-2014.) |
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Theorem | rmxfval 35798* | Value of the X sequence. Not used after rmxyval 35809 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.) |
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Theorem | rmyfval 35799* | Value of the Y sequence. Not used after rmxyval 35809 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.) |
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Theorem | rmspecsqrtnq 35800 | The discriminant used to define the X and Y sequences has an irrational square root. (Contributed by Stefan O'Rear, 21-Sep-2014.) |
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