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Using elementary complex arithmetic, it is easy to show that the sum or difference of a pair of inverse integral cotangents [A], [B] can be expressed as an inverse rational cotangent. Since [A] = arg (A + i) and [B] = arg (B + i), and since the sum of the arguments of two complex numbers is the argument of their product (because e^i(x+y) = e^ix . e^iy), then [A] + [B] = arg {(A + i).(B + i)} = arg {(AB-1) + i(A+B)} = [(AB-1) / (A+B)]. Similarly, [A] - [B] = [(AB+1) / (B-A)].
Conversely, "Dodgson's Formula" states that [X] = [X+a] + [X+b], or [X] = [b-X] - [X-a], where a.b = (X² + 1) (and b>X>a>0) - i.e. any two complementary factors of (X² + 1) can be used to obtain two "pair" representations of [X]. When X is an integer, and a and b are also integers, the members of these pairs will obviously also be inverse integral cotangents.
Dodgson's formula applies to rational, as well as integral, values of X. The next section discusses the case where X is rational, and it is required to obtain representations of [X] as sums or differences of inverse integral cotangents.
Consider [110443] - [4841182] = [34245479/303] (= [X]) = [113021 + (116/303)]. Is it possible to find any other representations of this inverse rational cotangent as the sum or difference of a pair of inverse integral cotangents?
To apply Dodgson's Formula, one must first calculate (X² + 1) = (34245479² + 303²)/303²
= 1172752832031250/303².
In this example, to ensure that integral cotangent values are obtained, 'a' and 'b' in Dodgson's Formula will be fractions whose denominator is 303, and whose numerators are complementary factors of 1172752832031250. It is therefore necessary to factorise this number completely. This is not too difficult. Because it is known that [110443] - [4841182] = 10[1]-17[2]+3[107], and since the primes 229 and 5 are associated with [107] and [2] respectively, it is to be expected that 229³, as well as a high power of 5, are factors. In fact, 1172752832031250 = 2.5^11.229³.
In order to obtain a pair of inverse integral cotangents equivalent to [X], one must find factors of 1172752832031250 which, because the fractional part of X is 116/303, are congruent (modulo 303) either to 116 or to 187 (= 303-116). In practice, as can easily be shown, if any factor is congruent to 116 or to 187, its complementary factor is also.
It is necessary to generate a factor table which shows, for each factor of 1172752832031250, the remainder when that factor is divided by 303. In this case, the easiest way to do this is to make a table with 4 rows and 12 columns; the 4 rows correspond to the factors 1, 229, 229² and 229³, and the 12 columns to 1, 5, 5², ...., 5^10 and 5^11. Each entry in the table is obtained by dividing the product of the row and column values (i.e. the actual factor) by 303 and entering the remainder (when generating the table it is easiest to work from left to right, using a pocket calculator, multiplying the entry in each column by 5 and if necessary subtracting multiples of 303 to obtain the entry to be entered in the adjacent column). The factor 2 can be omitted from this table, on the assumption that it is always present in the factors complementary to those in the table.
Factor table (modulo 303) for 1172752832031250 ('2' omitted)
1 5 5² 5³ 5^4 5^5 5^6 5^7 5^8 5^9 5^10 5^11
|------------------------------------------------------------
1 | 1 5 25 125 19 95 172 254 58 290 238 281
|
229 | 229 236 271 143 109 242 301 293 253 53 265 113
|
229² | 22 110 247 23 115 272 148 134 64 17 85 122
|
229³ | 190 41 205 116 277 173 259 83 112 257 73 62
There is only one eligible entry, that corresponding to the factor 5³.229³ (= 1501123625). Because the entry's value is 116, X (= 113021 + (116/303)) should be decremented by 1501123625/303, i.e. by 4954203 + (116/303), giving -4841182. The complementary factor is 2.5^8 (= 781250), and 781250/303 = 2578 + (116/303), so the other integral cotangent value is 113021 - 2578 = 110443.
Because the inverse rational cotangent [34245479/303] was originally derived from [110443] - [4841182], it was inevitable that there would be at least one eligible entry in its factor table (in 4., above). It transpired that this was the only eligible entry, implying that there are no other representations as a pair of inverse integral cotangents.
It may be possible to obtain a representation as a triplet of inverse integral cotangents. Because [113021] is the nearest integral value to [34245479/303], it will be instructive to see whether any such triplets can be obtained whose first value is [113021] - i.e. to determine the inverse rational cotangent which represents [113021] - [110443] + [4841182], and try to resolve this into one or more integral pairs.
The inverse rational cotangent is obtained from the product of (113021 + i) and (34245479 - 303i), which, after removal of a common factor 2, is (1935229141181 + 58i), so the cotangent value is 33366019675 + (31/58). The numerator value to be factorised in this case is 1935229141181² + 58² = 3745111828876150830078125.
[Note: This value is within the capacity of the MS Windows Calculator, which may be used to compute it, after which this large number can be stored in the Calculator's single memory - it will be needed in practice, after working out an "eligible" factor, to find the complementary factor. It will also be found useful to "memorise" the value 1935229141181 (for incrementing or decrementing) by using the Calculator's "Edit/Copy" facility, retrieving the value when required using "Edit/Paste".]
Before attempting to factorise 3745111828876150830078125 one does well to remember that (as in 3., above) a high power of 5, and 229³, will be among the factors. In addition, attempting to "reduce" [113021] shows that it is in fact irreducible, with the associated prime factor 4312541, a most helpful fact! The factors are actually 5^11.229³.1481.4312541.
This time the "modulo 58" factor table is most conveniently arranged with the same twelve columns as in 4. above, but with eight rows which correspond to the factors 1, 229, 229², 229³, 1481, 1481.229, 1481.229² and 1481.229³. The prime factor 4312541 is assumed only to participate in "complementary factors". The crucial entries to look for in the factor table this time are 31 and 27.
Factor table (modulo 58) for 3745111828876150830078125 ('4312541' omitted)
1 5 5² 5³ 5^4 5^5 5^6 5^7 5^8 5^9 5^10 5^11
|-----------------------------------------------------------
1 | 1 5 25 9 45 51 23 57 53 33 49 13
|
229 | 55 43 41 31 39 21 47 3 15 17 27 19
|
229² | 9 45 51 23 57 53 33 49 13 7 35 1
|
229³ | 31 39 21 47 3 15 17 27 19 37 11 55
|
1481 | 31 39 21 47 3 15 17 27 19 37 11 55
|
1481.229 | 23 57 53 33 49 13 7 35 1 5 25 9
|
1481.229² | 47 3 15 17 27 19 37 11 55 43 41 31
|
1481.229³ | 33 49 13 7 35 1 5 25 9 45 51 23
(Observe how, in this particular table, sequences of remainders repeat themselves in different rows, simplifying the task of generating the table).
There are eight eligible entries in the table; in four cases (31) the pairs of complementary factors will be subtracted from 1935229141181/58 to generate the integer pairs, in the other four cases (27) they will be added.
Taking as example the entry '31' at the beginning of the 4th row, the corresponding factor is 229³ = 12008989. The first fraction to be subtracted from 1935229141181/58 is therefore 12008989/58; the difference is 1935217132192/58 = 33365812624, so the first member of the pair is [33365812624]. Dividing 3745111828876150830078125 by 12008989 gives the complementary factor 311859043994140625; subtracting 311859043994140625/58 from 1935229141181/58 gives -311857108764999444/58 = -5376846702844818, so the other member of the pair is -[5376846702844818].
Thus, one solution (out of the eight available) is:
[113021] - [110443] + [4841182] = [33365812624] - [5376846702844818].
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