Changes and additions to Keith Matthews' BCMATH programs

4th November 2024. Added the 3x+3^k generalised Collatz mapping.
31st October 2024. Improved the printing of the output in 2-adic.html and p-adic.html and restricted the output to at most 1000 characters.
30th October 2024. Reinstated 2-adic.html.
15th December 2023. Fixed two variable misprints in binomial.
12th December 2023. binomial now returns the value of ⁿC_k.
29th June 2023. Replaced "cycles -1 → -1" by "cycles 7 → 7" in Example 4 of G. Venturini.
17th April 2023. Added missing cycles -1 → -1 and 4 → 4 to Example 4 of G. Venturini.
13th April 2023. Added a 6-branched mapping and associated conjecture.
4th April 2023. Added Example 4 of G. Venturini.
3rd April 2023. Added mapping m = 71 to mappings of Brian Gurbaxani.
31st March 2023. Added Example 5 of Venturini.
17th March 2023. Corrected a matrix Q_T(8) in The generalized 3x+1 mapping: George Leigh's approach.
15th March 2023. Added Combining the 5x+1 and 3x+1 mappings.
12th March 2023. Added bcwiggin.html.
9th March 2023. Added goncalves-greenfield-madrid1.html.
8th March 2023. Added goncalves-greenfield-madrid.html.
5th March 2023. More improvements to bell_numbers.html.
1st March 2023. More improvements to bell_numbers.html.
28th February 2023. Added bell_numbers.html.
26th February 2023. Fixed an error in alternating_numbers.html.
20th November 2022. Added alternating_numbers.html.
20th November 2022. Added secant_numbers.html.
19th September 2022. Added outline of algorithm to legendre.html and florida.html
15th September 2022. Improved the output for the Legendre and Florida transforms.
21st July 2022. Decreased the bounds for m and n in cfrac_nthroot.html.
6th March 2022. Changed directory name "pdfs" to "PDFS" in many places.
23rd February 2022. Fixed cut and paste error in 5th row of Q(10) in http://www.numbertheory.org/php/gurbaxani.html.
13th December 2021. Fixed cut and paste errors associated with powercomposite.html and ordercomposite.html.
20th September 2021. Updated florida.html and On solving a binary quadratic diophantine equation, K.R. Matthews, J.P. Robertson, Rocky Mountain Journal of Mathematics, 51(4): 1369-1385 (August 2021).
4th August 2021. Added note describing the algorithms in stoltvialmm.html and binaryformsfs.html.
31st July 2021. In binaryformsfs.html, I realised that the ((α_i – b β_i)/2a, β_i) are the fundamental solutions - so no need to examine their conjugates.
4th June 2021. Added two mappings of Brian Gurbaxani.
23rd March 2021. Added missing partial quotients from surd.html.
19th March 2021. Added solving the congruence ax² + bx + c ≡ 0 (mod n).
2nd January 2021. Added logabd.html (finding the continued fraction of log_ba/d).
16th December 2020. Added a statement about the cycles of a generalization of mapping 4 of Pruthviraj Hahari, where (k² - 1)x is replaced by k^t-2(k² - 1)x, t ≥ 2.
13th December 2020. Added a generalization of the first mapping of Pruthviraj Hajari.
12th December 2020. Added a generalization of the second mapping of Pruthviraj Hajari.
11th December 2020. Added a second mapping of Pruthviraj Hajari.
10th December 2020. Added a mapping of Pruthviraj Hajari.
8th August 2019. Programs log3.html and log4.html are now on the one page logab.html.
11th June 2019. Fixed a global variable problem in axbmodm.html.
12th December 2018. log4.html fails to live up to expectations when (a,b,m,n)=(991,2,146,148).
19th November 2018. Replaced log3.html by log4.html. We amazingly seem to get output that consists of correct partial quotients; but this awaits proof.
15th November 2018. Replaced log1.html by log3.html. We have simplified an earlier algorihtm and amazingly seem to get output that consists of correct partial quotients; but this awaits proof.
17th July 2018. Added barina8.html.
23rd April 2018. There have been some issues with log.html.
3rd April 2018. Replaced log.html by log1.html, the only change being that the parameter c is now taken to be b^r. This seems to give a more reliable output for the partial quotients.
14th December 2017. Fixed an error in the algorithm for finding SFSs in patzstolt.html.
12th December 2017. In surd.html, we now restrict the v in (u + t√d)/v to be positive.
14th November 2017. Improved the output of surd.html.
1st October 2017. Fixed a printing error in patz.html.
7th August 2017. Added a table of the first 250,000 partial quotients of the continued fraction expansion of e³ to davison.html.
27th July 2017. Added Solving ax² + bxy + cy² = n, where d = b² – 4ac > 0, d non-square, gcd(a,b,c)=1, where |n| < (√d)/2.
18th July 2017. Removed the cycle finding program for generalized 3x+1 mappings. Instead, please use the CALC version, as it gives the user more freedom to experiment with parameters.
17th July 2017. Added a 6 branch mapping.
29th May 2017. Added markov-forms-cycle.html.
15th May 2017. Added pell4.html.
10th April 2017. Program surd.html now prints the equation Q₀A_n-1² - 2P₀A_n-1B_n-1 + {(P₀²-D)/Q₀} B_n-1² = (-1)ⁿQ_n.
14th March 2017, Restored access to solving the general quadratic diophantine equation.
10th March 2017. Search range upperbound in aaa.html is now restricted to 1000000.
16th February 2017. Thanks to help from Wilberd van der Kallen, the LLLHNF pseudocode has been updated and now produces the correct HNF of a 6x5 matrix communicated by Benjamin Hildprecht.
7th February 2017. The LLL-based algorithm of Havas-Majewski and Matthews, for finding the Hermite normal form of an integer matrix, sometimes delivers a matrix which is not quite in Hermite normal form. This also is the case with the CALC version and is currently being investigated.
4th February 2017, Removed access to solving the general quadratic diophantine equation, as a bug is causing access problems to my site.
1st January 2017. Removed absolute values from the definition of A_i+1 in reduce.html, as well as all references to s_i.
29th July 2016. Fixed a coding error in partition.html.
21st July 2016. Changed old version of generalized Lu Pei mapping to Lu_Pei0.html.
18th July 2016. Fixed some errors in solving the general quadratic diophantine equation.
22nd June 2016. Added Calculating the product of two quadratic irrationals.
7th June 2016. Fixed a similar error to the previous one in patzstolt.html, this time in the function patz().
6th June 2016. Fixed an error in patzstolt.html which had caused faulty output when using the nonverbose option.
29th May 2016. Added Finding the least prime factor of an integer.
23rd May 2016. Added Conway's cycling sequence.
8th May 2016. Treated the cases d=5 and 13 in a better way in patzstolt.html.
15th April 2016. Fixed a minor sign error in the output of patzstolt.html.
24th March 2016. Changed it back again to its former state!
23rd March 2016. Changed the way I deal with D=13 in patzstolt.html.
22nd February 2016. I fixed another bug in patzstolt.html which showed up when D =4k+2.
19th February 2016. Fixed a bug in patzstolt.html
14th February 2016. Added patzstolt.html, which finds the Stolt fundamental solutions of x² -dy² = 4n, using ideas of John Robertson.
8th February 2016. Improved the output of patz.html.
13th January 2016. Fixed yet another small but significant error in patz_.php that caused havoc in binary.html and binarygen.html with the example 59x² - 16xy + y² = -4 and 59x² - 16xy + y² = -1.
29th December 2015. Fixed a bug in binary.html.
5th November 2015. Barry Smith found another error in Calculating the order of a binary indefinite quadratic form in the class group. This has now been fixed.
4th November 2015. Barry Smith found an error in Calculating the order of a binary indefinite quadratic form in the class group. The program had been returning h(d)! This has now been fixed.
15th October 2015. Fixed a bug in stolt_fundamental.html.
8th September 2015. Tightened up the code for finding the Stolt fundamental solutions. This had a flow-on effect for aaa.html.
7th September 2015. Corrected another error in aaa.html!
7th September 2015. Corrected error in the output of aaa.html, where in the case of even discriminant, I had U² – Dv² = 1, instead of U² – (D/4)v² = 1.
21st August 2015. I now list all Stolt-fundamental solutions in stolt.html, not just the positive ones.
13th June 2015. Added Solving a system of linear congruences with possibly differing moduli.
10th June 2015. Added Solving a system of linear diophantine congruences AX ≡ B (mod m) using the Smith normal form of an integer matrix.
25th May 2015. Added Expressing a prime 5n ± 1 in the form x² + xy – y², with x > 0, y > 0 using an algorithm of Christina Doran, Shen Lu and Barry R. Smith.
21st May 2015. Added Finding the modular inverse of m modulo n using an algorithm of Christina Doran, Shen Lu and Barry R. Smith.
18th May 2015. Corrected an error in the output of binarygen.html, namely replaced F/2 by F and G/2 by G in two places.
17th May 2015. Corrected radicand in the output when solving equations such as x² – 7y² = 109.
15th May 2015. Improved the output for solving the general quadratic diophantine equation by dividing through by the gcd of four coefficients on the first two lines of each solution.
5th May 2015. Improved the description of the algorithm behind solving the general quadratic diophantine equation.
3rd May 2015. Improved the output display of the general quadratic diophantine equation and improved the coding layout.
25th April 2015. The general quadratic diophantine equation is now solvable when gcd(a,b,c) > 1 and d = b² – 4ac > 0, d non-square.
18th April 2015. Added solving the general quadratic diophantine equation.
7th April 2015. Added the John Robertson method for solving ax² + bxy + cy² + dx + ey + f= 0, where d = b² – 4ac > 0, d non-square.
31st March 2015. Fixed an error in binarygen.html, when D=5 and an > 0.
24th March 2015. Fixed an error in the stolt program which had resulted in the wrong answer if N < 0 and v = √(4A|N|/D).
24th March 2015. Improved binarygen.html. It now gives formulae for the solutions in each class, instead of just a representative. I got the idea from Legendre's book Theorie des nombres.
15th March 2015. posrep.html now finds all solutions, not just primitive ones.
15th March 2015. Improved legendreneg.html.
15th March 2015. Improved ssw.html.
13th March 2015. I tightened up the last two changes, so as to deal with certain special cases.
12th March 2015. Added solving ax² + bxy + cy² + dx + ey + f = 0, where d = b² – 4ac < 0, a nonzero.
8th March 2015. Added solving ax² + bxy + cy² + dx + ey + f = 0, where d = b² – 4ac > 0, d non-square and a is nonzero.
7th March 2015. We now get all fundamental solutions (primitive and imprimitive) in the LMM algorithm.
7th March 2015. Improved the printing of the fundamental solutions in LMM algorithm.
21st February 2015. Added Wildberger's algorithm for solving Pell's equation.
2nd February 2015. Added lprimrootneg.html, lprimroot_generator.htm and lprimrootneg_generator.htm.
15th January 2015. Fixed an error in the Stolt fundamental solutions program.
13th January 2015. Fixed a double counting error in the Stolt fundamental solutions program.
17th December 2014. Improved the Stolt fundamental solutions program.
23rd November 2014. Added a second methof for testing Dujella's unicity conjecture.
22nd November 2014. Corrected a flaw in the code for Stolt fundamental solutions program.
16th November 2014. Improved the code for Stolt fundamental solutions program.
22nd October 2014. Added Solving the diophantine equation ax²+bxy+cy²=n, a > 0, n nonzero, b²-4ac > 0, gcd(a,b,c)=1, using the upper bounds of Bengt Stolt.
28th August 2014. Added upper bound for x in nagell_fundamental.html.
27th August 2014. Fixed an error in a variable name in Hermite-Serret program.
24th August 2014. Added Solving the diophantine equation ax²+bxy+cy²=n, b²-4ac < 0, gcd(a,b,c)=1, a > 0, n nonzero, using a continued fraction algorithm of Lagrange 1770.
24th July 2014. Corrected the printing of an error message and also improved the quality of the verbose mode for Solving a positive definite binary quadratic form diophantine equation.
15th July 2014. Added Solving the diophantine equation ax²+bxy+cy²=n, b²-4ac < 0, gcd(a,b,c)=1, a > 0, n nonzero, using a substitution of Lagrange 1770.
21st May 2014. Added Testing Dujella's unicity conjecture.
17th May 2014. Increased the number a from 3 to 5 in brent($n).
16th May 2014. Increased the number of primes used in Miller's test from 5 to 16.
28th March 2014. Added prime constellations.
26th February 2014. Added list of irregular primes p < 10000.
16th December 2013. Added patzpos.html.
4th December 2013. Added a second function to markov.html.
15th November 2013. Added listing of Markoff numbers in numerical order in markoff_triples.html.
14th November 2013. Added markoff_triples.html.
16th September 2013. Improved the description of rank.html.
8th September 2013. Further debugging and improving of rank.html.
5th September 2013. Added rank.html.
21st July 2013. Added Note on the Markoff numbers conjecture to stolt.html
19th July 2013. Added stolt.html.
17th July 2013. Removed a pair of minus signs in the output of the basis for non-negative solutions.
16th July 2013. Improved the coding of Frattini's algorithm in nagell_fundamental.html. Also added explanatory note.
14th July 2013. Fixed an error in the coding of Frattini's algorithm.
13th July 2013. Added Frattini's algorithm to nagell_fundamental.html.
28th June 2013. Fixed a mathematical error which led to the omission in nagell_fundamental.html of the possibility (0,y) as a fundamental solution of x²-Dy²=N when N < 0.
30th April 2013. Added calculating the minimal multipliers for m consecutive Fibonacci numbers.
4th February 2013. Changed the printing of ω in surd_construction.html so that it now is in standard form. Also created cfrac_.php which contains pn($a,$n), qn($a,$n), printcfrac($d,$t,$u,$v) and standardform($d,$t,$u,$v).
30th January 2013. Increased the upper limits for D and |N| in nagell_fundamental.
5th January 2013. Fixed a logical error in negative_pell.html and removed the dependence on variable t.
2nd January 2013. Added ternary.html.
30th December 2012. Added calculating the polynomial partial quotients for the continued fraction arising from an exceptional solution (k(t),x(t),y(t)) to Dujella's diophantine equation x² - (k²+1)y² = k².
22nd December 2012. Added calculating a continued fraction arising from an exceptional solution to Dujella's diophantine equation x² - (k²+1)y² = k².
21st December 2012. Added a restriction in nagell_fundamental, whereby the upper bound shall not exceed 10⁶.
14th December 2012. Added Nagell's method of solving x²-Dy²=N.
28th September 2012. Added an extension of the Hermite_Serret algorithm.
8th May 2012. Added solving ax²-by²=±1
19th March 2012. Noticed a bug in ./patz.html. I had used abs($n) instead of bcabs($n) in patz_.php.
9th January 2012. Added finding the divisors of a positive integer.
22nd November 2011. Fixed an error in computing the period of the least integer continued fraction expansion of a quadratic surd.
21st November 2011. Program for reducing an indefinite binary quadratic form using an algorithm of Don Zagier.
18th November 2011. Program for finding the least integer continued fraction expansion of a quadratic surd.
3rd November 2011. Program for solving AX=B in integers, now finds the shortest solutions.
2nd November 2011. Added finding the closest lattice points to a given point.
27th October 2011. Fixed a logical error in my OCF program that was pointed out to me by John Robertson.
26th October 2011. Continued fixing the previous error!
25th October 2011. Fixed a small but important error in axb.html, which was copied from the CALC version.
24th October 2011. Changed the logic of lllgcd. After initially finding a short multiplier using LLLGCD, it repeatedly replaces this multiplier by a shorter one until such a one does not exists and then exits with a list of all shortest multipliers.
18th October 2011. Updated lllgcd so that it now gets all shortest multiplier vectors for the extended gcd problem.
11th October 2011. Added finding the shortest vector in a lattice.
6th October 2011. Fixed inclusion file redundancies in euclid.html and ordercomposite.html.
28th September 2011. Updated LLL to allow the user to enter an α satisfying 1/4 < α ≤ 1.
27th September 2011. Updated MLLL to allow the user to enter an α satisfying 1/4 < α ≤ 1.
24th September 2011. Finding the Smith normal form of an integer matrix.
19th September 2011. Finding a LLL reduced basis for an integer matrix whose rows may be LD.
14th September 2011. Finding a LLL reduced basis for an integer matrix whose rows are LI.
12th September 2011. Finding a small solution of a system of linear diophantine equations AX=b using the Hermite normal form of an integer matrix and the Havas-Majewski-Matthews LLL-based algorithm.
11th September 2011. Replaced the initial version LLL based Hermite normal form algorithm by one with more input flexibility and which allows increased dimensions.
10th September 2011. Added the LLL based Hermite normal form algorithm.
5th September 2011. Added the case of finding the shortest multipliers for the extended gcd problem for four positive integers.
3rd September 2011. Added the case of finding the shortest multipliers for the extended gcd problem for three positive integers.
2nd September 2011. Added finding small multipliers for the extended gcd problem.
26th August 2011. Added Tomás Oliveira e Silva's mappings T₅ and T₇.
22nd August 2011. Added Geoffrey Caverney's function.
10th August 2011. Added extra stuff on F₃ to the Benoit Cloitre examples page.
8th August 2011. Added option for running F₂₉ on the Benoit Cloitre examples page.
3rd August 2011. Added option for running F₃ on the Benoit Cloitre examples page.
23rd July 2011. Added three mappings of Benoit Cloitre.
22nd July 2011. Added class number tables generated by the previous two programs.
22nd July 2011. Calculating the class number h(d) for binary quadratic forms of negative discriminant d now has alternatives for (i) fundamental, (ii) odd fundamental, (iii) even fundamental discriminants, as well as all discriminants.
21st July 2011. Calculating the class number h(d) for binary quadratic forms of positive discriminant d now has alternatives for (i) fundamental, (ii) odd fundamental, (iii) even fundamental discriminants, as well as all discriminants.
20th July 2011. Added Finding the nth power under composition of indefinite binary quadratic forms of positive discriminant.
20th July 2011. Added Finding the order of under composition of a binary indefinite quadratic forms of positive discriminant in the class group.
19th July 2011. Added composition of two indefinite binary quadratic forms of the same discriminant.
19th July 2011. Reducing a form of positive discriminant now has a nonverbose option.
18th July 2011. 11am Aust EST. I have reorganised the position of library.php amongst the various php files and this may have rendered some programs temporarily unworkable!
18th July 2011. Added order under composition of a positive definite quadratic form of negative discriminant in the class group.
15th July 2011. Added n-th power under composition of a binary positive definite quadratic form.
14th July 2011. Added composition of two binary positive definite quadratic forms of the same discriminant.
23rd June 2011. Added the partition number.
15th June 2011. Added the Kronecker symbol.
9th June 2011. Added Bernoulli numbers.
8th June 2011. Added Tangent numbers.
17th March 2011. Rewrote and simplifed code behind 2-adic square root.
15th March 2011. Rewrote and simplifed code behind p-adic square root, after a bug notification by Milton Minervino.
10th January 2011. Added Lu Pei's three-branched mapping conjecture.
28th December 2010. Added another interesting generalisation of the 3x-1 mapping arising from an example of Lu Pei (November 1997).
27th December 2010. Added an interesting generalisation of the 3x+1 mapping due to Walter Carnielli.
12th November 2010. Simplified the optimal cfrac program.
3rd November 2010. Added calculating the optimal continued fraction of √d for a range of consecutive d.
3rd November 2010. Added option of merely printing the continued fraction in optimal cfrac program.
5th September 2010. Added feature of finding all solutions y of φ(y) divides n to carmichael.html.
4th September 2010. Improved the logic of carmichael program - q[prime_count] now equals 0.
31st August 2010. Added tree diagram to carmichael.html.
26th August 2010. Added a program for solving φ(x)=n, where φ(x) is Euler's totient function.
22nd July 2010. Added Testing Mridul Sen's prime conjecture.
22nd July 2010. Added Generating Lucas pseudoprimes.
4th July 2010. Improved the code for initial_matrix($m,$n,$q) by using a suggestion of Alan Offer that I use the recurrence relations in his note.
30th June 2010. Got rid of the zero partial quotient in Finding the regular continued fraction of a rational multiple m/n of e^1/q.
24th June 2010. Added Finding the regular continued fraction of a rational multiple m/n of e^1/q.
24th June 2010. Added Another factorization of a 2x2 non-singular non-negative matrix.
18th June 2010. Added testing a quadratic surd for being NSCF-reduced. This is a different and more elegant test than a previous test, but does not produce the reduced NSCF predecessor.
28th May 2010. Replaced old nearest square continued fraction expansion by one which detects a reduced complete quotient using the Matthews-Robertson criterion.
19th May 2010. Added Example 8 of G. Venturini.
14th April 2010. Added Frequency (mod m) program for generalized 3x+1 mappings.
3rd January 2010. Added Fibonacci and Lucas numbers.
15th December 2009. Added testing a quadratic surd for being NSCF-reduced.
15th December 2009. Added testing a quadratic surd for being RCF-reduced.
17th November 2009. Replaced OCF program by faster version.
24th June 2009. Added Collatz type function of Mridul Sen.
29th March 2009. Added producing a quadratic surd equivalent to a given one.
17th March 2009. Added finding the positive and negative representations of a quadratic surd, as far as the end of the first period.
5th February 2009. Added listing the nearest square continued fraction of √d over a range of consecutive d.
5th February 2009. Added listing the nearest integer continued fraction of √d over a range of consecutive d.
5th February 2009. Added listing the simple continued fractionof √d over a range of consecutive d.
8th August 2008. Added an algorithm for solving x²-xy-(D-1)/4y²=±1 using the nearest square continued fraction of (1+√D)/2, D≡ 1(mod 4).
7th August 2008. Added an algorithm for solving Pell's equation using the nearest square continued fraction algorithm (NSCF) and midpoint criteria.
7th August 2008. Added an algorithm for solving Pell's equation using the nearest integer continued fraction algorithm from Perron's book (NICF-P) and midpoint criteria.
28th April 2008. Added an algorithm for converting the RCF, up to the end of the first period, into the NICF (removed 18th August 2008).
18th April 2008. Added Selenius' algorithm for converting the RCF up to the end of the first period to the NSCF (removed 18th August 2008).
4th April 2008. Added the optimal continued fraction algorithm for rationals.
2nd April 2008. Added the choice of not printing the b_k and u_k / v_k in the optimal continued fraction algorithm.
14th February 2008. Added the optimal continued fraction algorithm of Wieb Bosma.
2nd November 2007. Added nearest square continued fraction expansion of a quadratic rational.
2nd November 2007. Added nearest square Pell equation algorithm.
19th October 2007. Added nearest integer continued fraction expansion of a quadratic rational
18th October 2007. Added a second NICF expansion of m/n to nearest integer Euclidean algorithm.
11th October 2007. Added nearest integer Pell equation algorithm.
5th October 2007. Added nearest integer Euclidean algorithm.
13th February 2007. Added computing the period of the continued fraction expansion of √d.
9th February 2007. Added decimal to rational program.
7th February 2007. Added calculating the power of a quadratic irrational.
7th December 2006. Replaced continued fraction-based reduction algorithm for indefinite binary quadratic forms by Henri Cohen's reduction algorithm.
5th December 2006. Cleared up some remaining confusion regarding the reduction algorithm for indefinite binary quadratic forms. We now expand 1/τ and consequently make changes F_i = (-1)ⁱt/f_i and the unimodular transformation becomes x = Y, y = -X + (-1)ⁱtg_iY.
Cleared up some underlying confusion regarding the reduction algorithm for indefinite binary quadratic forms. We now find a unimodular matrix that transforms (a,b,c) to reduced form.
29th September 2006. Added testing the solubility of the negative Pell equation.
29th September 2006. Added the Grytczuk, Luca, Wojtowicz construction of non-square d such that the continued fraction of √d has odd length period, testing over ranges for u and v.
19th September 2006. Added finding the gcd of n positive integers.
19th September 2006. Added Calculating the quadratic irrationality whose simple continued fraction is given.
6th September 2006. Added the constructive version of the Aubry-Thue theorem.
28th August 2006. Added the Grytczuk, Luca, Wojtowicz construction of non-square d such that the continued fraction of √d has odd length period.
16th August 2006. Added σ_k(n) and λ(n).
20th July 2006. Added finding the continued fraction of the unique positive root of a trinomial axⁿ+bx^n-1+c.
20th July 2006. Added finding the continued fraction of the unique positive root of a trinomial axⁿ+bx+c.
5th June 2006. Added perfect-power testing.
2nd June 2006. Added finding the simple continued fraction of the n-th root of a positive rational.
27th May 2006. Notice that the following BCMATH program shows that one has to use the bccomp() function in conditionals:
```
<?php
$test1="100000000000001415402032694799700273500898";
$test2="100000000000001421659378396209449729633403";
if($test1 > $test2){
   echo "$test1 > $test2";
}
if($test1 == $test2){
   echo "$test1 = $test2";
}
if($test1 < $test2){
   echo "$test1 < $test2";
}
?>
```
The program produces output
100000000000001415402032694799700273500898 = 100000000000001421659378396209449729633403
I will now have to start the tedious task of changing back to using bccomp().
26th May 2006. Added Representing an integer by a positive definite binary quadratic form.
20th May 2006. Added the option of printing the convergents in Calculating the simple continued fraction of a quadratic irrational.
3rd May 2006. Noticed that the Fox-Landi program is exiting prematurely and doesn't declare state 3 to be transient for the matrix.

1 0 0

0 1 0

1 1 1

This was due to double-counting and was easily fixed.
The bug was also present in Generalized 3x+1 functions and Markov matrices.
29th April 2006. Added finding the cyclic classes and cyclic form to the periodic matrix program.
10th March 2006. Fixed the error in Fox-Landi. It was due to my confusion in only adjoining elements i_k+1,..,i_r to S_{i_k}, instead of taking the unions of S_{i_k+1},..,S_{i_r} with S_{i_k}.
9th March 2006. Fox-Landi is playing up again in relation to analysing Q(m)! eg. The program hangs when d=2,m=8,m[0]=1,m[1]=3,x[0]=0,x[1]=1.
25th February 2006. Added option of viewing transitions between classes in George Leigh's program.
25th February 2006. Added cycle-finding program for generalized 3x+1 functions.
24th February 2006. Added Example 7 of G. Venturini.
23rd February 2006. Added Example 1 of G. Venturini.
23rd February 2006. 3pm. I believe I have fixed the flaws in Fox-Landi. The error affected both programs for processing Q(m). There still remains an inefficiency due to my initial confusion about communicating transient states.
22nd February 2006. Found an error in Fox-Landi. I am currently working on it.
22nd February 2006. Added a program for finding the transition matrix of George Leigh's Markov chain Y_n(x).
9th January 2006. Added finding the canonical form of the Markov matrix Q(m) arising from generalized 3x+1 functions.
6th January 2006. Added a verbose option to Fox_Landi program.
4th January 2006. Fixed an error in the construction of the permutation of rows and columns which gives the canonical form in the Fox-Landi program.
29th December 2005. Added the Fox-Landi algorithm. I'm still learning about _REQUEST, _SESSION etc. so there may be some bugs in the front end of this program.
14th June 2005. After an email from a user of binary.html, I realised I had absent-mindedly forgotten to print " no solution" in the case that an examinaton of the relevant cfracs revealed this to be the case in binary.html.
25th January 2005. Added padiclog.html. (Coded August 2004)
9th September 2004. Added the option of printing the complete quotients in surd.html.
8th August 2004. Added binary.html.
5th August 2004. Found a transcription error in squareroot_.php. Also wrote a better version of quadratic.html.
16th July 2004. Added patz.html.
15th July 2004. Fixed some bugs in squareroot_.php, which corresponded to bugs in BC program squareroot. These would have affected cornacchia.
21st May 2004. Added discrete_log.
21st May 2004. While adding cong and cong1 to library.php, I noticed a bug in finding squareroots (mod m). It was in a call to cong() instead of cong1 in sqroot1 in the file squareroot_.php.
21st May 2004. Adding more functions to library.php. This requires care to avoid clashes with other include files.
20th May 2004. More carelessness. Changed name of include file mpower.php to library.php in arithmeticm.php.
28th April 2004. Improved my use of include files by creating a large include file called library.php.
27th April 2004. Fixed a small global variable problem arising when calculating Ramanujan's tau function tau_composite(n). Having factored n and obtained global variables representing the prime factors q[i] and their exponents k[i], I inadvertently changed these on subsequent factorisations involved in calculating tau(q[i]).
22nd April 2004. Hopefully I've fixed problems concerning multiple declarations in include files.
22nd April 2004. Fixed a small printing bug in square roots (mod n), when omega(n)=1.
22nd April 2004. Started introducing some order in the way I used include files. This has been causing some errors.
21st April 2004. Added calculating Ramanujan's tau function.
16th April 2004. Added solving ax²+by²=m.
16th April 2004. Added finding square roots (mod n).
7th April 2004. Added Pollard p-1 to factoring program.
6th April 2004. Added ord_ma.
6th April 2004. Added least primitive root (mod p).
6th April 2004. Added factoring and computing omega(n), phi(n), d(n), sigma(n) and mu(n).
24th March 2004. Added finding partial quotients of e^l/m.
22nd March 2004. Added factorising a non-negative non-singular matrix.
19th March 2004. Inserted a missing sign term in p-adic_.php.
17th March 2004. Added p-adic square root.
11th March 2004. Added 2adic square root.
27th February 2004. Added prime-power factorization of n!
17th July 2003. Improved the output of the log program when log_ba is rational.
27th February 2004. Added prime-power factorization of binomial coefficients.
17th July 2003. Improved the output of the log program when log_ba is rational.
13th June 2003. Fixed similar error to below in calculating classnumbers of binary quadratic forms of positive discriminant.
11th June 2003. Fixed an error in calculating class numbers of real quadratic fields.

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