- Added
**root3**. This contains`cycle(x,y,n)`. - Improved code and output for
`surd(d,t,u,v,e)`in**patz**. - Removed a print statement in
**exceptionals**which was causing a problem. - 18th July 2016. Fixed some errors in
`patz`in function`sswgeneral(a,b,c,d,e,f)`. - 1st July 2016. (i) I had forgotten to include
`lineareqn(a,b,c,d,e,f)`and`lineareqnn(a,b,c,d,r,s,n`to file**patz**. (ii) In`lineareqn(a,b,c,d,e,f)`I had written`delta=ad-bc`instead of`delta=a*d-b*c`. - 24th May 2016. Added
`conwaysequence1(a,b,n)`to**phi**. - 23rd May 2016. Added
`conwaycycles(a,b)`to**phi**. - 10th September 2015. Added
`stolt0(d,n,flag)`to**stolt**. - 8th September 2015. Programs
`aa1(a,b,c)`and`aa2(a,b,c,n)`under construction. - 26th August 2015. Updated
**patz**so as to include`aa1(a,b,c)`and`aa2(a,b,c,n)`. - 25th May 2015. Added
`rep(p)`. - 21st May 2015. Added
`inverse(m,n)`. - 13th May 2015. I realized that I had two different functions called
`hyperbola`- one in**patz**, the other in**powerdd**. So I've renamed the first one`hyperboladd`. - 7th May 2015. Updated
**patz**which now contains`sswgeneral`,`hyperbola(a,b,c,d,e,f,dd,gg)`,`hyperbolagg(a,b,c,d,e,f,dd,gg)`and`ellipse(a,b,c,k,alpha,beta)`. It also involved updating**gcd**,**powerdd**,**genfacs**,**posformrep**,**reduceneg**and constructing the files**powertest3**and**congruence**. - 7th April 2015. Replaced
`c`by`globalsqc`and`count`by`globalsqcount`in**squareroot**and**squareroot2**. - 5th April 2015. Added
`ssw4`to**patz**. - 24th March 2015. Added to
**patz**. - 17th March 2015. In
`binary0(a,b,c,n,e,f)`, the case a=n=1 was dealt with at the start without using continued fractions. - 16th March 2015. Added functions
`binarygen(a,b,c,n)`and`binarygenlist(a,b,c,n)`to**patz**. - 11th March 2015. Added functions
`posrepgen(a,b,c,n), posrepgenlist(a,b,c,n), ssw1(a,b,c,d,e,f)`to**posformrep**. - 10th March 2015. Noticed a subtle bug in function
`quadratic(a,b,c,flag) when there was no soltion`. - 9th March 2015. Added functions
`patzgen(d,n)`and`patzgentest(d,n)`to**patz**. Also added`ssw0(a,b,c,d,e,f)`to**nagell_test**. - 7th March 2015. Corrected the signs of the fundamental solutions.
- 5th March 2015. I noticed that
**nagell_test**somehow became corrupted. This has now been fixed. - 21st February 2015. Added
**wildberger1**. - 2nd February 2015. Added
`lprimrootneg(p), lprimrootposmn(m,n), lprimrootnegmn(m,n),`to**phi**. Added`lucasnonverbose(n)`to**lucas**. - 15th January 2015. Corrected error introduced in previous change in
`fund()`in**stolt**. - 14th January 2015. Treated the case d=x
^{2}+ 1 separately at the start of`fund()`in**stolt**. - 14th January 2015. Fixed an error in
`fund>`which appeared when d=41. - 13th January 2015. Fixed a double counting error in
**stolt**when v=0. - 17th December 2014. Improved code in
**stolt**. - 22nd November 2014. Fixed faulty code in
**stolt**. - 20th November 2014. Added
**dujella_minus**. - 22nd October 2014. Added
**stolt**. - 20th August 2014. Added
**euclid0**. - 14th July 2014. Added
**posformrep**. - 14th May 2014. Increased the number of primes used in Miller's test to 17 in
**factors**and**phi**. - 28th March 2014. Made some improvements to
**primepatterns**. - 26th March 2014. Added
**primepatterns**. - 16th December 2013. Added
**patzpos**. - 13th November 2013. Added
**markoff_triples**. - 19th July 2013. Fixed a logical error in stolt(d,n) in the determination of ambiguous solution.
- 18th July 2013. Added stolt(d,n) to
**nagell_test**. - 14th July 2013. Now print two digits in the Nagell and Frattini bounds.
- 11th July 2013. Fixed an error in
`frattini(d,n)`in**nagell_test**. - 15th December 2012. Added
**nagell_test**. - 18th September 2012. Added
**squareroot2**. - 8th May 2012. Added
**pellab**. - 9th January 2012. Added
**genfacs**. - 27th October 2011. Deleted the OCF program due to a missing case.
- 5th August 2011. Added
**cloitrem**. - 20th July 2011. Added
**cloitre**. - 15th July 2011. Added
`ordercomposite`to**reduceneg**. - 15th July 2011. In
**classnoneg**, table(m,n) now calculates h(-d) for 3 ≤ m ≤ d ≤ n , d ≡ 0 or 3 (mod 4). - 14th July 2011. Added
`compose`,`compose1`,`power_compose`and`power_compose1`to**reduceneg**. - 14th July 2011. Added
`bezout`and`bezout1`to**gcd**. - 13th July 2011. Fixed an error in
**reduceneg**. - 24th June 2011. Reinstated
**powerdd**and tidied up things. - 23rd June 2011. Added
**partition**. - 15th June 2011. Added
**kronecker**and altered**jacobi**so that now`jacobi(m,1)=1`. - 9th June 2011. Added
**bernoulli**. - 8th June 2011. Added
**tangent**. - 17th March 2011. Updated
**padic**. This now contains simplified versions of`padic`and`twoadic`. - 15th March 2011. Replaced
`p-adic`by`padic`. Currently it does not contain the 2-adic squareroot algorithm, but this will soon be reinstated. - 26th November 2010. Forgot to add
`gcd()`and`gcd3()`to the new version of**nscf_pell**. - 25th November 2010. Replaced
**nscf_pell**. The new version uses a test for NSCF reduced which is better than the actual definition, which was used in the original program. - 25th November 2010. Brian Gladman has pointed out that in
**nscf_pell**, inside the function nscf_pqa(), the line d=d*(t*t) should be outside the braces. This error was also present in**nipell**. - 2nd November 2010. In
**ocf**, fixed a bug in`ocf_pqd1`, where I should have tested if(pk==3 && qk==3 && d==5) instead of if(pk==3 && qk==3). Also updated the ocf file by adding the function`ocf_pqd2`, which replaces`ocf_pqd1`. - 4th September 2010. Improved the logic in
**carmichael**. - 27th August 2010. Altered the printing in
**carmichael**. It now tests Carmichael's conjecture. - 26th August 2010. Added
**carmichael**. This necessicated updating**nprime**(to get a non-verbose version) and**lucas**(to deal with the possibility that n could be even). - 4th January 2010. Added
**fibonacci**. - 20th November 2009. Added
**ocf**. - 29th March 2009. Added
**equivalent**. - 3rd October 2008. Added
**spiral**. - 21st August 2008. Noticed that variable
`a`in**nipell**should have been declared to be auto. - 10th January 2008. Added
`nicf_pqa0(d,t,p,q,e)`to**nipell**. Bugs fixed subsequently. - 8th January 2008. I now convert the input surd (u +t√d)/v to
*standard form*in files**nipell**and**nscf_pell**. ie. (U +√D)/V, where gcd(U, V, (D - U^{2})/V) = 1.

- 1st December 2007. Removed auto variables t1 and t2 in nscf_pell(d,e). Apparently 26 is the maximum number of auto variables allowed in a BC function! Also tightened the logic in the definition of the partial numerators a[i].
- 15th November 2007. Added gcd3(a,b,c).
- Altered the code for the nearest square algorithm so as to agree with lines 5-6, page 25, of the A.A.K. Ayyangar paper, when |Q'| = |Q"|.
- 2nd November 2007. Added
`nscf_pqa(d,t,p,q,e)`to**nscf_pell**. - 1st November 2007.Made some small improvements and corrections (regarding the period length) to
**nscf_pell**and**nipell**. - 31st October 2007. Added
**nscf_pell**. - 22nd October 2007. Added
`nicf_pqa(d,t,u,v,e)`to**nipell**. - 18th October 2007. Added
`nicf(m,n)`to**lra**. - 12th October 2007. Corrected output about period in
**nipell**. - 11th October 2007. Added
**nipell**. - 12th February 2007. Added
**sqrtd_period**. - 9th February 2007. Added
**decimal2rational**. - 8th February 2007. Removed the restriction gcd(b,n)=1 from the
**decimal**program. - 15th November 2006. Small improvement to
**reducepos**. - 13th June 2006. Fixed a serious error in
**nprimeap**. - 8th August 2004. Added
`binary`to**patz**. - 4th August 2004. Added
`quadratic`to**squareroot**. - 16th July 2004. Added
**patz**. - 14th July 2004.
**squareroot**continues to haunt me.- I fixed a bug in relation to solving x
^{2}0 (mod 2). - Also had an incorrect construction for extending the solutions to original modulus n, so as to list them in the form ±P (mod n), with 0 ≤ P ≤ n/2.
- Replaced
`if(qglobal[i]==2 && number==1)`by`if(number==1)`. This error was giving the wrong answer in x^{2}0 (mod 6).

- I fixed a bug in relation to solving x
- 14th May 2004. Added
**forster_log**. - 13th May 2004. Noticed that I'd omitted a restriction that in
**discrete_log**, we must have p < 2^{32}-2^{16}in order to satisy BC array upper bound length of 2^{16}-1. - 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]).
- 21st April 2004. Added
`sigmak(k,n)`and`tau`to**phi**. - 15th April 2004. Added
**squareroot**, which is a cleaner version of**sqroot**. The new program contains`cornacchia(a,b,m)`. - 7th April 2004. Improved
**factor**and**phi**by incorporating`pollard`. - 24th March 2004. Changed name of
**cfrace**to**davison**. - 23rd March 2004. Added
**cfrace**. - 22nd March 2004. Added
**raney**. - 19th March 2004. Inserted a sign term in function
`padic`. - 17th March 2004. Changed name of
**2adic**to**p-adic**. Also added`padic(a,p,n)`. - 10th March 2004. Added
**2adic**. - 27th February 2004. Added
**factorial**. - 26th February 2004. Added
**binomial**. - 22nd May 2003. Added
**unimodular**. - 15th May 2003. Added the solution of equations x
^{2}-d*y^{2}=±3 and ±4 to**pell**. - 14th May 2003. Added
`table(m,n)`to**classnopos**. - 14th May 2003. Added
`table(m,n)`to**classnoneg**. - 12th May 2003. Fixed a small error in the function
`period`. It was supposed to be returning the cycle-length, not`global_count`! - 12th May 2003. Added
`class_number0(d)`to**classnopos**. - 7th May 2003. Fixed an error on lines 58-60 of
**classnoneg**. - 5th May 2003. Added
**classnopos**. - 1st May 2003. Added
**reduceneg**. - 29th April 2003. Added
**classnoneg**. - 28th April 2003. Changed the output of
**surd**to allow choice of printing complete quotients. - 27th April 2003. Added
**reducepos**. - 20th April 2003. Fixed a trivial printing error at the end of
**unit**. - 19th February 2003. Added program
**john**. - 23rd September 2002. Fixed errors on lines 104 and 112 of
**lagrange**, where I had 2 instead of n. - 18th September 2002.
**lagrange**now needs**sturm**.`gcdpi`and`sturm`now have an extra parameter e, where e=0 suppresses printing.`lagrange`has also been tightened up. We now check that the input polynomial satisfies the appropriate restrictions. Also the program now handles the case when the root above 1 is in fact rational. - 16th September 2002. Improved
`printp`in**sturm**. - 15th September 2002. Added
**sturm**. - 11th September 2002. Interchanged lines 43 and 44 in
**decimal**to fix a printing error. - 9th September 2002. Added
**nprime**and**nprimeap**. - 3rd September 2002.
**primes**now generates more primes. - 3rd September 2002. Made a small improvement to
**primes**when m=2. - 2nd September 2002. Added missing
`else b=0`at line 214 of**proth**. - 27th August 2002. Rewrote
`leastqnr`so that it ony needs`mpower`as in**gcd**. - 25th August 2002. Improved the output of
**surd**. - 22nd August 2002. Improved the output of
**pell**. - 9th August 2002. Added
**base**and**perfect_power**. - 31st July 2002. Replaced "<=" by "<" on line 11 of
`mthrootr()`in**gcd**. - 25th June 2002. Replaced all log programs by
**log**. - 14th June 2002. Added another log program -
**log2**. - 8th March 2002. I removed the line
`j=2`from`gcd1`in**gcd**and**sqroot**. (Kindly pointed out by Kjell Wooding.) - 10th August 2001. I reinstated
**log1**. - 24th July 2001. I replaced the above
**log**program by the one that is currently in my manuscript log.pdf. - 14th June 2001. Today I noticed problems with degenerate cases arising from small r in my
`log`algorithm. - 23rd February 2001. Added
**tomas2**. - 13th February 2001. Added
**tomas1**. - 20th October 2000. Altered
`unit`so that when d=5 it returned the correct answer. - 7th August 2000. Altered
`log(a,b,r,e)`to`log(a,b,d,r,e)` - 29th July 2000. Improved the printing in
**log**. - 6th May 2000. Fixed a bug in
**surd**. - 1st May 2000. Corrected printing of output in
**thue**: changed s to t at appropriate places. - 26th April 2000.
**surd**now prints P_{n}and Q_{n}, where the nth complete quotient is (P_{n}+sqrt(d))/Q_{n}. - 11th April 2000. Realised
**sqroot**still had some bugs. Hopefully these are now fixed. - 7th April 2000. Replaced
**sqroot**by an improved version which solves the general congruence x^{2}d (mod n). Any bug reports are gratefully received. - 6th April 2000. Realised
**sqroot**has some bugs. - 5th April 2000. Changed
`thue(d,p)`to`thue(d,u,p)`. - 4th April 2000. Added
**sqroot**. This finds all solutions of x^{2}d (mod n) where gcd(d,n)=1. - 3rd April 2000. Changed name of
`chineseab()`to`chineseb()`. Rewrote`chinesea()`and`chineseb()`. - 2nd April 2000. Added printing of partial quotients and truncated decimal expansion to
**log**. Removed printing of A[i]. Also added**thue**. Here d>1, is not a square, p an odd prime not dividing d-1. Then`thue(d,p)`finds x,y such that x^{2}-dy^{2}=kp, with small k. - 1st April 2000.
**jacobi**now does not need**gcd**and is a stand-alone program.

- 28th March 2000. Discovered that the old version of
**log**did not always give the correct partial quotients. The current version seems to be more reliable. Type`log(3,2,2,10,0)`to output roughly 20 partial quotients of log_{2}(3), whereas typing`log(3,2,2,10,1)`prints the A[i] of algorithm 1 of manuscript log.pdf. The number of partial quotients is returned in both cases. - 13th March 2000. Made some improvements to the output of
**log**. - 4th March 2000. Added
**rootd_modn**. This is a crude program for solving x^{2}d (mod n) with 0 ≤ x ≤ n/2 for small n. - Added
**log**. This is a discrete variant of Shank's algorithm for computing the simple continued fraction for log_{b}n, (Math. Tables and Aids to Computation. 8, 1954, 60-64), originally written by Alan Offer and recently improved by vacation scholar Sean Seefried and myself. It should currently be treated with caution. For while we believe that`log(n,b,z,e)`computes approximately 2z partial quotients of log_{b}n, we cannot guarantee the correctness of output!. -
**lagrange**was improved by Sean Seefried, using the binary search method in page 261 of*Number Theory with Computer Applications*, by R. Kumanduri and C. Romero, 17th December 1999. - Improved the
**euclid**program. It now prints the s[k] and t[k] associated with Euclid's algorithm. - Forgot to test possibility n
_{p}=2 in`leastqnr(p)`. Fixed 28th September 1999. - Cosmetic alterations to the
**tonelli**program, 24th August 1999. - Added
**leastqnr**, 13th August 1999. Also removed dependence of**proth**on**phi**and**gcd**.

- Removed dependence of
**lucas**on**phi**.

- Tightened up the
**discrete_log**program, 11th August 1999. - Added the
**discrete_log**program, 12th July 1999. - Added the
**tonelli**program, 29th June 1999. - Fixed bug in
**gcd**in lcma(m[ ],n) (should have returned b[n-1], not b[n]) 11th November 1998. - Added some functions to
**phi**that are normally in**gcd**, 9th October 1998. - Added
**recursion**, 3rd May 1998. - Added the
**lupei**program. 24th March 1998. - Added to
**lra**. 21st December 1997. - Added the
**lagrange**program. 17th December 1997. - Added the
**convergents**program. 15th December 1997. - Added the
**lra**program. 24th January 1997. - Added the
**venturini1**program. 1st April 1996. - Added a missing case at line 27 in
`unit(n)`. 5th November 1995. - Added the
**mordell**program. 26th September 1995. - Added the
**challenge**program. 27th August 1995. - Added the
**3branch**program. 26th August 1995. Fixed a bug there on 27th August 1995 - 15th June 95. Deleted the for loop in function
`b(n)`on line 253 of**factors**.

*Email*
*http://www.numbertheory.org/keith.html*