# RogerBW's Blog

 The Weekly Challenge 167: Circling the Gamma 02 June 2022 I’ve been doing the Weekly Challenges. The latest involved circular primes and the Lanczos approximation. (Note that this is open until 5 June 2022.) Task 1: Circular Prime Write a script to find out first 10 circular primes having at least 3 digits (base 10). A circular prime is a prime number with the property that the number generated at each intermediate step when cyclically permuting its (base 10) digits will also be prime. There's a trick here. Yes, 113 is a circular prime - but so are 131 and 311. What's wanted, according to the example, is just the lowest number in each group. The Sieve of Eratosthenes relies heavily on having a limit - it will generate all primes up to that limit with reasonable efficiency, but it doesn't then have a way of using that list to generate more primes. So I break this down by digit count: I'll look for all the 3-digit circular primes, then all the 4-digits, etc., until I have enough. (Eventually I may develop an extensible version of the sieve code, but not yet.) First the cyclic permutation. In all languages, I want to get these out as numbers, but stringifying the input and manipulating that seems like the best bet. I realised after I'd written this in all the languages that it would have been better to double up the string representation rather than mutating it, then take substrings, thus: ``````"123" → "123123" → "[123]123", "1[231]23", "12[312]3" `````` but never mind. Python: ``````def cyclicpermute(n): ss = str(n) o = [] for p in range(len(ss)): ss = ss[1:] + ss[0] o.append(int(ss)) return o `````` Then the actual search. ``````def circular(mindigits,ct): o = [] `````` Initialise the base value with the minimum digits (mindigits = 3 gives a base of 100). `````` base = 1 for p in range(mindigits-1): base = base * 10 while len(o) < ct: `````` For that number of digits, generate all the primes within it. Put these in a sorted list, and in a set that I'll use later. `````` pr = genprimes(base * 10) prs = set(pr) `````` Iterate through the list (dropping ones that are lower than the current `base`, which we wastefully calculate multiple times). `````` for cp in pr: if cp >= base: `````` Generate the permutations, and if any of them isn't prime (doesn't appear in `prs`) then drop out and carry on with the next in the list. `````` v = True cpp = cyclicpermute(cp) for cpc in cpp: if cpc not in prs: v = False break if v: `````` But if all the permutations are prime: put it on the output list and exit if we've got enough values. Then mark each of the values in the permutation as non-prime - so that, having done 113, when we meet 131 we won't also log that one. `````` o.append(cp) if len(o) >= ct: break for cpc in cpp: prs.discard(cpc) `````` If we've done all the n-digit primes without getting enough results, add a digit and start again. `````` base = base * 10 return o `````` Task 2: Gamma Function Implement subroutine gamma() using the Lanczos approximation method. Which basically needs complex numbers. I avoid the languages which don't have easily-accessible complex number libraries (so there's no JavaScript, Kotlin, Lua or PostScript implementation here). Ruby has one, but produces very inaccurate results; I'm not sure what's going on there. I use the algorithm given (in Python) at Wikipedia, but here's the Perl version (using Math::Complex). Some globals: ``````my \$EPSILON = 1e-7; my @p = (676.5203681218851, -1259.1392167224028, 771.32342877765313, -176.61502916214059, 12.507343278686905, -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7); `````` Dropping the imaginary part: ``````sub drop_imag(\$z0) { my \$z = \$z0; if (abs(Im(\$z0)) <= \$EPSILON) { \$z = Math::Complex->make(Re(\$z),0); } return \$z; } `````` The full algorithm using the global coefficients: ``````sub gamma(\$z0) { my \$z = Math::Complex->make(\$z0,0); my \$y; if (Re(\$z) < 0.5) { \$y = pi / (sin(pi * \$z) * gamma(1-\$z)); } else { \$z--; my \$x = 0.99999999999980993; foreach my \$i (0..\$#p) { \$x += \$p[\$i] / (\$z + \$i + 1); } my \$t = \$z + scalar(@p) - 0.5; \$y = sqrt(2 * pi) * \$t ** (\$z + 0.5) * exp(-\$t) * \$x; } return drop_imag(\$y); } `````` Full code on github. Comments on this post are now closed. If you have particular grounds for adding a late comment, comment on a more recent post quoting the URL of this one. 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