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How to find generators of a finite field Fp[x]/f(x) with f(x) is a irreducible polynomial over Fp.
Input: p (prime number), n (positive number), f (irreducible polynomial)
Output: g (generator)
I have p = 2, n =3, f = x^3 + x + 1
I am a newbie so I don't know where to start.
Do you have any solution? Plese help me step by step
To find a generator (primitive element) α(x) of a field GF(p^n), start with α(x) = x + 0, then try higher values until a primitive element α(x) is found.
For smaller fields, a brute force test to verify that powers of α(x) will generate every non-zero number of a field can be done.
cnt = 0
m = 1
do
cnt = cnt + 1
m = (m*α)%f(x)
while (m != 1)
if cnt == (p^n-1) then α(x) is a generator for GF(p^n).
For a faster approach with larger fields, find all prime factors of p^n-1. Let q = any of those prime factors. If α(x) is a generator for GF(p^n), then while operating in GF(p^n):
α(x)^(p^n-1) % f(x) == 1
α(x)^((p^n-1)/q) % f(x) != 1, for all q that are prime factors of p^n-1
In this case GF(2^3) is a 3 bit field and since 2^3-1 = 7, which is prime, then it's just two tests, shown in hex: x^3 + x + 1 = b (hex)
α(x)^7 % b == 1
α(x)^1 % b != 1
α(x) can be any of {2,3,4,5,6,7} = {x,x+1,x^2,...,x^2+x+1}
As another example, consider GF(2^4), f(x) = x^4 + x^3 + x^2 + x + 1 (hex 1f). The prime factors of 2^4-1 = 15 are 3 and 5, and 15/3 = 5 and 15/5 = 3. So the three tests are:
α(x)^f % 1f == 1
α(x)^5 % 1f != 1
α(x)^3 % 1f != 1
α(x) can be any of {3,5,6,7,9,a,b,e}
For larger fields, finding all prime factors of p^n-1 requires special algorithms and big number math. Wolfram alpha can handle up to around 2^128-1:
https://www.wolframalpha.com/input/?i=factor%282%5E64-1%29
This web page can factor large numbers and includes an explanation and source code:
https://www.alpertron.com.ar/ECM.HTM
To test for α(x)^(large number) = 1 or != 1, use exponentiation by repeated squaring while performing the math in GF(p^n).
https://en.wikipedia.org/wiki/Exponentiation_by_squaring
For large fields, where p^n is greater than 2^32 (4 billion), a primitive polynomial where α(x) = x is searched for, using the test mentioned above.
I want to find (num * (pow(b, p) - 1) / den) % mod. I know about binary exponentiation. But we can't do it straightforward. It is guaranteed that the numerator is divisible by the denominator. That means
[num * (pow(b, p) - 1)] % den == 0
constraints on mod: are 1 <= mod <= 10 ^ 9 and mod might be prime or composite
constraints on b: 1 <= b <= 10
constraints on p: 1 <= p <= (10^18)
constraints on num: 1 <= num <= (10^9)
constraints on den: 1 <= den <= (10^9)
Here pow(b, p) means b raised to power p(b ^ p). It is guaranteed that the numerator is divisible by the denominator. How can I do it with binary exponentiation
Your expression should rewritten to simplIfy it. First let k=num/den, with k integer according to your question.
So you have to compute
(k×(b^p-1))mod m=( (k mod m) × ((b^p -1) mod m) ) mod m
= ( (k mod m) × ( (b^p mod m) -1 mod m ) mod m ) mod m
= ((k mod m) × ((b^p mod m) + m-1) mod m) mod m (1)
So the real problem is to compute b^p mod m
Many languages (python, java, etc) already have a modular exponentiation in their standard libraries. Consult the documentation and use it. Otherwise, here is a C implementation.
unsigned long long modexp(unsigned long long b, unsigned long long e, unsigned long long m) {
if (m==1) return 0;
unsigned long long res=1;
unsigned long long bb = b % m;
while (e) {
if (e & 1)
res = (res*b) % m;
e >>= 1;
bb = (bb*bb) % m;
}
return res;
}
The implementation uses long long to fit your constraints. It relies on the classical trick of binary exponentiation. All values of b^l, where l is a power of two (l=2^t) are computed and stored in var bb and if the corresponding tth bit of e is set, this value of b^l is integrated in the result. Bit testing is done by checking the successive parities of e, while shifting e rightward at each step.
Last, the fact that (a×b)mod m=((a mod m)×(b mod m))mod m is used to avoid computation on very large numbers. We always have res<m and bb<m and hence res and bb are codable on standard integers.
Then you just have to apply (1) to get the final result.
EDIT according to the precisions given in the comments
To compute n=(3^p-1)/2 mod m, one can remark that
(3^p-1)/2 = x*m + n (as 3^p-1 is even, x is an integer, 0≤n<m)
3^p-1=x*2*m+2n (0≤2n<2m)
so 2n=(3^p-1) mod 2m
We can just apply the previous method with a modulo of 2*m, and divide the result (that will be even) by 2.
I was trying to implement various forms of queries on Hailstone Sequence.
Hailstone sequences are sequences of positive integers with the following properties:
1 is considered the terminating value for a sequence.
For any even positive integer i, the value that comes after i in the sequence is i/2.
For any odd positive integer j > 1, the value that comes after j in the sequence is 3j+1
Queries can be
hailSequence(Seed,Sequence): where the Sequence is the hailstone sequence generated from the given Seed.
hailStone(M,N): where N is the number that follows M in a hailstone sequence. E.g. if M is 5 then N should be 16, if M is 20 then N should be 10, etc.
hailStorm(Seed,Depth,HailTree): where HailTree is the tree of values that could preceed Seed in a sequence of the specified depth.
Example:
| ?- hailStorm(1,4,H).
H = hs(1,hs(2,hs(4,hs(8)))) ?
yes
| ?- hailStorm(5,3,H).
H = hs(5,hs(10,hs(3),hs(20))) ?
yes
Pictorial Representation
Now I've implemented the first two predicates:
hailSequence(1,[1]) :- !.
hailSequence(N,[N|S]) :- 0 is N mod 2, N1 is round(N / 2), hailSequence(N1,S).
hailSequence(N,[N|S]) :- 1 is N mod 2, N1 is (3 * N) + 1, hailSequence(N1, S).
hailStone(X,Y) :- nonvar(X), 0 is X mod 2, Y is round(X / 2).
hailStone(X,Y) :- nonvar(X), 1 is X mod 2, Y is (3 * X) + 1.
hailStone(X,Y) :- nonvar(Y), 1 is Y mod 3, T is round( (Y - 1) / 3), 1 is T mod 2, X is T.
For the hailStorm/2 predicate, I've written the following code, but it is not working as expected:
make_hs1(S,hs(S)).
make_hs2(S,R,hs(S,make_hs1(R,_))).
make_hs3(S,L,R,hs(S,make_hs1(L,_),make_hs1(R,_))).
hailStorm(S,1,hs(S)) :- !.
hailStorm(S,D,H) :- nonvar(S), nonvar(D), 4 is S mod 6, S=\= 4, make_hs3(S,hailStorm(round((S-1)/3),D-1,_X),hailStorm(2*S,D-1,_Y),H).
hailStorm(S,D,H) :- nonvar(S), nonvar(D), make_hs2(S,hailStorm(2*S,D-1,_X),H).
Output:
| ?- hailStorm(5,2,H).
H = hs(5,make_hs1(hailStorm(2*5,2-1,_),_))
yes
which is not the desired output,i.e.,
H = hs(5,hs(10)) ?
There are several issues expressed in the problem statement:
In Prolog, there are predicates and terms but not functions. Thinking of them as functions leads one to believe you can write terms such as, foo(bar(3), X*2)) and expect that Prolog will call bar(3) and evaluate X*2 and then pass these results as the two arguments to foo. But what Prolog does is pass these just as terms as you see them (actually, X*2 internally is the term, *(X,2)). And if bar(3) were called, it doesn't return a value, but rather either succeeds or fails.
is/2 is not a variable assignment operator, but rather an arithmetic expression evaluator. It evaluates the expression in the second argument and unifies it with the variable or atom on the left. It succeeds if it can unify and fails otherwise.
Although expressions such as 0 is N mod 2, N1 is round(N / 2) will work, you can take more advantage of integer arithmetic in Prolog and write it more appropriately as, 0 =:= N mod 2, N1 is N // 2 where =:= is the arithmetic comparison operator. You can also use bit operations: N /\ 1 =:= 0, N1 is N // 2.
You haven't defined a consistent definition for what a hail storm tree looks like. For example, sometimes your hs term has one argument, and sometimes it has three. This will lead to unification failures if you don't explicitly sort it out in your predicate hailStorm.
So your hailSequence is otherwise correct, but you don't need the cut. I would refactor it a little as:
hail_sequence(1, [1]).
hail_sequence(Seed, [Seed|Seq]) :-
Seed > 1,
Seed /\ 1 =:= 0,
S is Seed // 2,
hail_sequence(S, Seq).
hail_sequence(Seed, [Seed|Seq]) :-
Seed > 1,
Seed /\ 1 =:= 1,
S is Seed * 3 + 1,
hail_sequence(S, Seq).
Or more compactly, using a Prolog if-else pattern:
hail_sequence(1, [1]).
hail_sequence(Seed, [Seed|Seq]) :-
Seed > 1,
( Seed /\ 1 =:= 0
-> S is Seed // 2
; S is Seed * 3 + 1
),
hail_sequence(S, Seq).
Your description for hailStone doesn't say it needs to be "bidirectional" but your implementation implies that's what you wanted. As such, it appears incomplete since it's missing the case:
hailStone(X, Y) :- nonvar(Y), Y is X * 2.
I would refactor this using a little CLPFD since it will give the "bidirectionality" without having to check var and nonvar. I'm also going to distinguish hail_stone1 and hail_stone2 for reasons you'll see later. These represent the two ways in which a hail stone can be generated.
hail_stone(S, P) :-
hail_stone1(S, P) ; hail_stone2(S, P).
hail_stone1(S, P) :-
S #> 1,
0 #= S rem 2,
P #= S // 2.
hail_stone2(S, P) :-
S #> 1,
1 #= S rem 2,
P #= S * 3 + 1.
Note that S must be constrained to be > 1 since there is no hail stone after 1. If you want these using var and nonvar, I'll leave that as an exercise to convert back. :)
Now to the sequence. First, I would make a clean definition of what a tree looks like. Since it's a binary tree, the common representation would be:
hs(N, Left, Right)
Where Left and Right are branchs (sub-trees), which could have the value nul, n, nil or whatever other atom you wish to represent an empty tree. Now we have a consistent, 3-argument term to represent the tree.
Then the predicate can be more easily defined to yield a hail storm:
hail_storm(S, 1, hs(S, nil, nil)). % Depth of 1
hail_storm(S, N, hs(S, HSL, HSR)) :-
N > 1,
N1 is N - 1,
% Left branch will be the first hail stone sequence method
( hail_stone1(S1, S) % there may not be a sequence match
-> hail_storm(S1, N1, HSL)
; HSL = nil
),
% Right branch will be the second hail stone sequence method
( hail_stone2(S2, S) % there may not be a sequence match
-> hail_storm(S2, N1, HSR)
; HSR = nil
).
From which we get, for example:
| ?- hail_storm(10, 4, Storm).
Storm = hs(10,hs(20,hs(40,hs(80,nil,nil),hs(13,nil,nil)),nil),hs(3,hs(6,hs(12,nil,nil),nil),nil)) ? ;
(1 ms) no
If you want to use the less symmetrical and, arguably, less canonical definition of binary tree:
hs(N) % leaf node
hs(N, S) % one sub tree
hs(N, L, R) % two sub trees
Then the hail_storm/3 predicate becomes slightly more complex but manageable:
hail_storm(S, 1, hs(S)).
hail_storm(S, N, HS) :-
N > 1,
N1 is N - 1,
( hail_stone1(S1, S)
-> hail_storm(S1, N1, HSL),
( hail_stone2(S2, S)
-> hail_storm(S2, N1, HSR),
HS = hs(S, HSL, HSR)
; HS = hs(S, HSL)
)
; ( hail_stone2(S2, S)
-> hail_storm(S2, N1, HSR),
HS = hs(S, HSR)
; HS = hs(S)
)
).
From which we get:
| ?- hail_storm1(10, 4, Storm).
Storm = hs(10,hs(20,hs(40,hs(80),hs(13))),hs(3,hs(6,hs(12)))) ? ;
no
I'm having trouble figuring out the difference between the two. Say you have these givens:
a[0] = 10
a[1] = 13
a[2] = 17
a[3] = 19
x = 0
y = 3
OPERATOR PRECEDENCE:
++, --
*, /, % Left Associative
+, - Left Associative
OPERAND EVALUATION ORDER:
Right to Left
Given the rules above, how would I evaluate the expression below?
a[++x] + ++x % 7 % y
According to my professor, the answer is 18, but I cannot figure out why. From what I understand associativity is the order same precedence operators are evaluated and operand evaluation order is the order operands get evaluated so something like 2 % 7 would be 2 with left to right operand evaluation order and 1 with operation evaluation order. Can anyone explain how my professor got the answer of 18?
The precedence and associativity tell you how the expression is (implicitly) parenthesised. The evaluation order then determines in which order the subexpressions are evaluated.
Let us look at the example:
a[++x] + ++x % 7 % y
On the top level, there are + and % as operators. + has lower precedence, so that's
a[++x] + (++x % 7 % y)
The right subexpression has two %, and that is left associative, hence
a[++x] + ((++x % 7) % y)
Now with right-to-left evaluation order, ((++x % 7) % y) is evaluated first. Again with right-to-left evaluation order, y is evaluated first, resulting in 3. Then ++x % 7 is evaluated. First 7, then ++x. The latter results in 1. So that's 1 % 7 = 1. I'll leave the rest to you, since it's homework.
You have () + () % 7 % y. Based on the rules, () % 7 is evaluated before ... % y and that before () + ....
In ++x % 7 you first evaluate ++x and get 1 and x=1. 1 % 7 = 1.
Then you do 1 % y or 1 % 3 and get 1.
Now you do a[++x] + 1. Remembering that x=1, you get a[2] + 1 = 17 + 1 = 18.
Is there an algorithm that can calculate the digits of a repeating-decimal ratio without starting at the beginning?
I'm looking for a solution that doesn't use arbitrarily sized integers, since this should work for cases where the decimal expansion may be arbitrarily long.
For example, 33/59 expands to a repeating decimal with 58 digits. If I wanted to verify that, how could I calculate the digits starting at the 58th place?
Edited - with the ratio 2124679 / 2147483647, how to get the hundred digits in the 2147484600th through 2147484700th places.
OK, 3rd try's a charm :)
I can't believe I forgot about modular exponentiation.
So to steal/summarize from my 2nd answer, the nth digit of x/y is the 1st digit of (10n-1x mod y)/y = floor(10 * (10n-1x mod y) / y) mod 10.
The part that takes all the time is the 10n-1 mod y, but we can do that with fast (O(log n)) modular exponentiation. With this in place, it's not worth trying to do the cycle-finding algorithm.
However, you do need the ability to do (a * b mod y) where a and b are numbers that may be as large as y. (if y requires 32 bits, then you need to do 32x32 multiply and then 64-bit % 32-bit modulus, or you need an algorithm that circumvents this limitation. See my listing that follows, since I ran into this limitation with Javascript.)
So here's a new version.
function abmody(a,b,y)
{
var x = 0;
// binary fun here
while (a > 0)
{
if (a & 1)
x = (x + b) % y;
b = (2 * b) % y;
a >>>= 1;
}
return x;
}
function digits2(x,y,n1,n2)
{
// the nth digit of x/y = floor(10 * (10^(n-1)*x mod y) / y) mod 10.
var m = n1-1;
var A = 1, B = 10;
while (m > 0)
{
// loop invariant: 10^(n1-1) = A*(B^m) mod y
if (m & 1)
{
// A = (A * B) % y but javascript doesn't have enough sig. digits
A = abmody(A,B,y);
}
// B = (B * B) % y but javascript doesn't have enough sig. digits
B = abmody(B,B,y);
m >>>= 1;
}
x = x % y;
// A = (A * x) % y;
A = abmody(A,x,y);
var answer = "";
for (var i = n1; i <= n2; ++i)
{
var digit = Math.floor(10*A/y)%10;
answer += digit;
A = (A * 10) % y;
}
return answer;
}
(You'll note that the structures of abmody() and the modular exponentiation are the same; both are based on Russian peasant multiplication.)
And results:
js>digits2(2124679,214748367,214748300,214748400)
20513882650385881630475914166090026658968726872786883636698387559799232373208220950057329190307649696
js>digits2(122222,990000,100,110)
65656565656
js>digits2(1,7,1,7)
1428571
js>digits2(1,7,601,607)
1428571
js>digits2(2124679,2147483647,2147484600,2147484700)
04837181235122113132440537741612893408915444001981729642479554583541841517920532039329657349423345806
edit: (I'm leaving post here for posterity. But please don't upvote it anymore: it may be theoretically useful but it's not really practical. I have posted another answer which is much more useful from a practical point of view, doesn't require any factoring, and doesn't require the use of bignums.)
#Daniel Bruckner has the right approach, I think. (with a few additional twists required)
Maybe there's a simpler method, but the following will always work:
Let's use the examples q = x/y = 33/57820 and 44/65 in addition to 33/59, for reasons that may become clear shortly.
Step 1: Factor the denominator (specifically factor out 2's and 5's)
Write q = x/y = x/(2a25a5z). Factors of 2 and 5 in the denominator do not cause repeated decimals. So the remaining factor z is coprime to 10. In fact, the next step requires factoring z, so you might as well factor the whole thing.
Calculate a10 = max(a2, a5) which is the smallest exponent of 10 that is a multiple of the factors of 2 and 5 in y.
In our example 57820 = 2 * 2 * 5 * 7 * 7 * 59, so a2 = 2, a5 = 1, a10 = 2, z = 7 * 7 * 59 = 2891.
In our example 33/59, 59 is a prime and contains no factors of 2 or 5, so a2 = a5 = a10 = 0.
In our example 44/65, 65 = 5*13, and a2 = 0, a5 = a10 = 1.
Just for reference I found a good online factoring calculator here. (even does totients which is important for the next step)
Step 2: Use Euler's Theorem or Carmichael's Theorem.
What we want is a number n such that 10n - 1 is divisible by z, or in other words, 10n ≡ 1 mod z. Euler's function φ(z) and Carmichael's function λ(z) will both give you valid values for n, with λ(z) giving you the smaller number and φ(z) being perhaps a little easier to calculate. This isn't too hard, it just means factoring z and doing a little math.
φ(2891) = 7 * 6 * 58 = 2436
λ(2891) = lcm(7*6, 58) = 1218
This means that 102436 ≡ 101218 ≡ 1 (mod 2891).
For the simpler fraction 33/59, φ(59) = λ(59) = 58, so 1058 ≡ 1 (mod 59).
For 44/65 = 44/(5*13), φ(13) = λ(13) = 12.
So what? Well, the period of the repeating decimal must divide both φ(z) and λ(z), so they effectively give you upper bounds on the period of the repeating decimal.
Step 3: More number crunching
Let's use n = λ(z). If we subtract Q' = 10a10x/y from Q'' = 10(a10 + n)x/y, we get:
m = 10a10(10n - 1)x/y
which is an integer because 10a10 is a multiple of the factors of 2 and 5 of y, and 10n-1 is a multiple of the remaining factors of y.
What we've done here is to shift left the original number q by a10 places to get Q', and shift left q by a10 + n places to get Q'', which are repeating decimals, but the difference between them is an integer we can calculate.
Then we can rewrite x/y as m / 10a10 / (10n - 1).
Consider the example q = 44/65 = 44/(5*13)
a10 = 1, and λ(13) = 12, so Q' = 101q and Q'' = 1012+1q.
m = Q'' - Q' = (1012 - 1) * 101 * (44/65) = 153846153846*44 = 6769230769224
so q = 6769230769224 / 10 / (1012 - 1).
The other fractions 33/57820 and 33/59 lead to larger fractions.
Step 4: Find the nonrepeating and repeating decimal parts.
Notice that for k between 1 and 9, k/9 = 0.kkkkkkkkkkkkk...
Similarly note that a 2-digit number kl between 1 and 99, k/99 = 0.klklklklklkl...
This generalizes: for k-digit patterns abc...ij, this number abc...ij/(10k-1) = 0.abc...ijabc...ijabc...ij...
If you follow the pattern, you'll see that what we have to do is to take this (potentially) huge integer m we got in the previous step, and write it as m = s*(10n-1) + r, where 1 ≤ r < 10n-1.
This leads to the final answer:
s is the non-repeating part
r is the repeating part (zero-padded on the left if necessary to ensure that it is n digits)
with a10 =
0, the decimal point is between the
nonrepeating and repeating part; if
a10 > 0 then it is located
a10 places to the left of
the junction between s and r.
For 44/65, we get 6769230769224 = 6 * (1012-1) + 769230769230
s = 6, r = 769230769230, and 44/65 = 0.6769230769230 where the underline here designates the repeated part.
You can make the numbers smaller by finding the smallest value of n in step 2, by starting with the Carmichael function λ(z) and seeing if any of its factors lead to values of n such that 10n ≡ 1 (mod z).
update: For the curious, the Python interpeter seems to be the easiest way to calculate with bignums. (pow(x,y) calculates xy, and // and % are integer division and remainder, respectively.) Here's an example:
>>> N = pow(10,12)-1
>>> m = N*pow(10,1)*44//65
>>> m
6769230769224
>>> r=m%N
>>> r
769230769230
>>> s=m//N
>>> s
6
>>> 44/65
0.67692307692307696
>>> N = pow(10,58)-1
>>> m=N*33//59
>>> m
5593220338983050847457627118644067796610169491525423728813
>>> r=m%N
>>> r
5593220338983050847457627118644067796610169491525423728813
>>> s=m//N
>>> s
0
>>> 33/59
0.55932203389830504
>>> N = pow(10,1218)-1
>>> m = N*pow(10,2)*33//57820
>>> m
57073676928398478035281909373919059149083362158422691110342442061570390868211691
45624351435489450017295053614666205465236942234520927014873746108612936700103770
32168799723279142165340712556208924247665167762020062262193012798339674852992044
27533725354548599100657212037357315807679003804911795226565202352127291594603943
27222414389484607402282947077135939121411276374956762365963334486336907644413697
68246281563472846765824974057419578000691802144586648218609477689380837080594949
84434451746800415081286751988931165686613628502248356969906606710480802490487720
51193358699411968177101349014181943964026288481494292632307160152196471809062608
09408509166378415773088896575579384296091317883085437564856451054998270494638533
37945347630577654790729851262538913870632998962296783120027672085783465928744379
10757523348322379799377378069872016603251470079557246627464545140089934278796264
26841923209961950882047734347976478727084053960567277758561051539259771705292286
40608785887236250432376340366655136630923555863023175371843652715323417502594258
04219993081978554133517813905223106191629194050501556554825319958491871324801106
88343133863714977516430300933932895191975095122794880664130058803182289865098581
80560359737115185
>>> r=m%N
>>> r
57073676928398478035281909373919059149083362158422691110342442061570390868211691
45624351435489450017295053614666205465236942234520927014873746108612936700103770
32168799723279142165340712556208924247665167762020062262193012798339674852992044
27533725354548599100657212037357315807679003804911795226565202352127291594603943
27222414389484607402282947077135939121411276374956762365963334486336907644413697
68246281563472846765824974057419578000691802144586648218609477689380837080594949
84434451746800415081286751988931165686613628502248356969906606710480802490487720
51193358699411968177101349014181943964026288481494292632307160152196471809062608
09408509166378415773088896575579384296091317883085437564856451054998270494638533
37945347630577654790729851262538913870632998962296783120027672085783465928744379
10757523348322379799377378069872016603251470079557246627464545140089934278796264
26841923209961950882047734347976478727084053960567277758561051539259771705292286
40608785887236250432376340366655136630923555863023175371843652715323417502594258
04219993081978554133517813905223106191629194050501556554825319958491871324801106
88343133863714977516430300933932895191975095122794880664130058803182289865098581
80560359737115185
>>> s=m//N
>>> s
0
>>> 33/57820
0.00057073676928398479
with the overloaded Python % string operator usable for zero-padding, to see the full set of repeated digits:
>>> "%01218d" % r
'0570736769283984780352819093739190591490833621584226911103424420615703908682116
91456243514354894500172950536146662054652369422345209270148737461086129367001037
70321687997232791421653407125562089242476651677620200622621930127983396748529920
44275337253545485991006572120373573158076790038049117952265652023521272915946039
43272224143894846074022829470771359391214112763749567623659633344863369076444136
97682462815634728467658249740574195780006918021445866482186094776893808370805949
49844344517468004150812867519889311656866136285022483569699066067104808024904877
20511933586994119681771013490141819439640262884814942926323071601521964718090626
08094085091663784157730888965755793842960913178830854375648564510549982704946385
33379453476305776547907298512625389138706329989622967831200276720857834659287443
79107575233483223797993773780698720166032514700795572466274645451400899342787962
64268419232099619508820477343479764787270840539605672777585610515392597717052922
86406087858872362504323763403666551366309235558630231753718436527153234175025942
58042199930819785541335178139052231061916291940505015565548253199584918713248011
06883431338637149775164303009339328951919750951227948806641300588031822898650985
8180560359737115185'
As a general technique, rational fractions have a non-repeating part followed by a repeating part, like this:
nnn.xxxxxxxxrrrrrr
xxxxxxxx is the nonrepeating part and rrrrrr is the repeating part.
Determine the length of the nonrepeating part.
If the digit in question is in the nonrepeating part, then calculate it directly using division.
If the digit in question is in the repeating part, calculate its position within the repeating sequence (you now know the lengths of everything), and pick out the correct digit.
The above is a rough outline and would need more precision to implement in an actual algorithm, but it should get you started.
AHA! caffiend: your comment to my other (longer) answer (specifically "duplicate remainders") leads me to a very simple solution that is O(n) where n = the sum of the lengths of the nonrepeating + repeating parts, and requires only integer math with numbers between 0 and 10*y where y is the denominator.
Here's a Javascript function to get the nth digit to the right of the decimal point for the rational number x/y:
function digit(x,y,n)
{
if (n == 0)
return Math.floor(x/y)%10;
return digit(10*(x%y),y,n-1);
}
It's recursive rather than iterative, and is not smart enough to detect cycles (the 10000th digit of 1/3 is obviously 3, but this keeps on going until it reaches the 10000th iteration), but it works at least until the stack runs out of memory.
Basically this works because of two facts:
the nth digit of x/y is the (n-1)th digit of 10x/y (example: the 6th digit of 1/7 is the 5th digit of 10/7 is the 4th digit of 100/7 etc.)
the nth digit of x/y is the nth digit of (x%y)/y (example: the 5th digit of 10/7 is also the 5th digit of 3/7)
We can tweak this to be an iterative routine and combine it with Floyd's cycle-finding algorithm (which I learned as the "rho" method from a Martin Gardner column) to get something that shortcuts this approach.
Here's a javascript function that computes a solution with this approach:
function digit(x,y,n,returnstruct)
{
function kernel(x,y) { return 10*(x%y); }
var period = 0;
var x1 = x;
var x2 = x;
var i = 0;
while (n > 0)
{
n--;
i++;
x1 = kernel(x1,y); // iterate once
x2 = kernel(x2,y);
x2 = kernel(x2,y); // iterate twice
// have both 1x and 2x iterations reached the same state?
if (x1 == x2)
{
period = i;
n = n % period;
i = 0;
// start again in case the nonrepeating part gave us a
// multiple of the period rather than the period itself
}
}
var answer=Math.floor(x1/y);
if (returnstruct)
return {period: period, digit: answer,
toString: function()
{
return 'period='+this.period+',digit='+this.digit;
}};
else
return answer;
}
And an example of running the nth digit of 1/700:
js>1/700
0.0014285714285714286
js>n=10000000
10000000
js>rs=digit(1,700,n,true)
period=6,digit=4
js>n%6
4
js>rs=digit(1,700,4,true)
period=0,digit=4
Same thing for 33/59:
js>33/59
0.559322033898305
js>rs=digit(33,59,3,true)
period=0,digit=9
js>rs=digit(33,59,61,true)
period=58,digit=9
js>rs=digit(33,59,61+58,true)
period=58,digit=9
And 122222/990000 (long nonrepeating part):
js>122222/990000
0.12345656565656565
js>digit(122222,990000,5,true)
period=0,digit=5
js>digit(122222,990000,7,true)
period=6,digit=5
js>digit(122222,990000,9,true)
period=2,digit=5
js>digit(122222,990000,9999,true)
period=2,digit=5
js>digit(122222,990000,10000,true)
period=2,digit=6
Here's another function that finds a stretch of digits:
// find digits n1 through n2 of x/y
function digits(x,y,n1,n2,returnstruct)
{
function kernel(x,y) { return 10*(x%y); }
var period = 0;
var x1 = x;
var x2 = x;
var i = 0;
var answer='';
while (n2 >= 0)
{
// time to print out digits?
if (n1 <= 0)
answer = answer + Math.floor(x1/y);
n1--,n2--;
i++;
x1 = kernel(x1,y); // iterate once
x2 = kernel(x2,y);
x2 = kernel(x2,y); // iterate twice
// have both 1x and 2x iterations reached the same state?
if (x1 == x2)
{
period = i;
if (n1 > period)
{
var jumpahead = n1 - (n1 % period);
n1 -= jumpahead, n2 -= jumpahead;
}
i = 0;
// start again in case the nonrepeating part gave us a
// multiple of the period rather than the period itself
}
}
if (returnstruct)
return {period: period, digits: answer,
toString: function()
{
return 'period='+this.period+',digits='+this.digits;
}};
else
return answer;
}
I've included the results for your answer (assuming that Javascript #'s didn't overflow):
js>digit(1,7,1,7,true)
period=6,digits=1428571
js>digit(1,7,601,607,true)
period=6,digits=1428571
js>1/7
0.14285714285714285
js>digit(2124679,214748367,214748300,214748400,true)
period=1759780,digits=20513882650385881630475914166090026658968726872786883636698387559799232373208220950057329190307649696
js>digit(122222,990000,100,110,true)
period=2,digits=65656565656
Ad hoc I have no good idea. Maybe continued fractions can help. I am going to think a bit about it ...
UPDATE
From Fermat's little theorem and because 39 is prime the following holds. (= indicates congruence)
10^39 = 10 (39)
Because 10 is coprime to 39.
10^(39 - 1) = 1 (39)
10^38 - 1 = 0 (39)
[to be continued tomorow]
I was to tiered to recognize that 39 is not prime ... ^^ I am going to update and the answer in the next days and present the whole idea. Thanks for noting that 39 is not prime.
The short answer for a/b with a < b and an assumed period length p ...
calculate k = (10^p - 1) / b and verify that it is an integer, else a/b has not a period of p
calculate c = k * a
convert c to its decimal represenation and left pad it with zeros to a total length of p
the i-th digit after the decimal point is the (i mod p)-th digit of the paded decimal representation (i = 0 is the first digit after the decimal point - we are developers)
Example
a = 3
b = 7
p = 6
k = (10^6 - 1) / 7
= 142,857
c = 142,857 * 3
= 428,571
Padding is not required and we conclude.
3 ______
- = 0.428571
7