Разложение дроби в сумму аликвотных |
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Не будем в этой статье, повторять всеми продублированные и растиражированные истины, откуда эти дроби появились и почему в Древнем Египте не могли додуматься до обычных дробей.
Аликвотная дробь, дробь которая имеет в числителе единицу. Например
\(\cfrac{1}{44}\),\(\cfrac{1}{121981}\),\(\cfrac{1}{3}\)
Так как их начали применять в Древнем Египте, то их и назвали египетскими.
Любую дробь, можно представить множеством вариантов сумм аликвотных дробей.
Например
Есть также много разных алгоритмов разложения произвольной дроби в сумму египетских.
Мы остановимся на алгоритме, который описал/открыл Голомб
Но прежде чем начнем давайте определимся как например решать дробь вида \(\cfrac{2}{p}\)
Тривиальное решение как \(\cfrac{1}{p}+\cfrac{1}{p}=\cfrac{2}{p}\) не рассматриваем.
Считаем, что каждое слагаемое в сумме должно быть уникальным.
В Древнем Египте, под эту дробь был исписан пергамент, то есть достаточно дорогой материал потратили на запись дробей вида \(\cfrac{2}{p}\), что влечет подозрения, что это была важная веха в математике тех времен, и наверняка египтяне потратили немало времени, что бы разложить подобные дроби.
С сожалению, они бы сэкономили, если бы знали что такую дробь, всегда можно разложить на сумму трех аликвотных дробей.
\(\cfrac{2}{p}=\cfrac{1}{p}+\cfrac{1}{(p+1)}+\cfrac{1}{p(p+1)}\)
И даже двух аликвотных дробей, если \( p \) нечетное число
\(\cfrac{2}{p}=\cfrac{(1+\cfrac{1}{p})}{\cfrac{p+1}{2}}\)
\(\cfrac{2}{17}=\cfrac{(1+\cfrac{1}{17})}{9}\)
Не долго думая, можно легко догадаться что дробь \(\cfrac{3}{p}\) можно получить как
\(\cfrac{3}{p}=\cfrac{1}{p}+\cfrac{2}{(p+1)}+\cfrac{2}{p(p+1)}\)
и в общем случае
\(\cfrac{n}{p}=\cfrac{1}{p}+\cfrac{n-1}{(p+1)}+\cfrac{n-1}{p(p+1)}\)
Конечно, скажете Вы, какие же тут аликвотные дроби, если в числителе не единицы!
Совершенно верно, но данная формула показывает, когда заданную дробь, абсолютно точно можно разложить на три аликвотных дроби.
Если \(\cfrac{p+1}{n-1}\) делится нацело без остатка, то данная дробь раскладывается на сумму трех египетских дробей.
Поэтому, когда Вам на уроке зададут задачу разложить дробь \cfrac{17}{303} то, посчитав, что \(\cfrac{303+1}{17-1}=19\), кричите и поднимайте руку, утверждая что её можно разложить на три аликвотные дроби
\(\cfrac{17}{303}=\cfrac{1}{303}+\cfrac{1}{19}+\cfrac{1}{19*303}\)
Оптимальное ли такое разложение? Нет конечно. Её можно представить в виде суммы двух дробей, но для этого надо нам надо погрузится в рассмотрение различных алгоритмов. И мы это сделаем позже.
Теперь что касается формулы
\(\cfrac{2}{p}=\cfrac{(1+\cfrac{1}{p})}{\cfrac{p+1}{2}}\)
Её тоже можно обобщить и написать что
\(\cfrac{n}{p}=\cfrac{(1+\cfrac{1}{p})}{\cfrac{p+1}{n}}\)
Вывод: дробь раскладывается на сумму двух аликвотных дробей если \(\cfrac{p+1}{n}\) делится без остатка.
Является ли это обязательным условием? Нет конечно. Например вышеупомянутая дробь \(\cfrac{17}{303}\) под это условие не попадает но тем не менее
\(\cfrac{17}{303}=\cfrac{1}{18}+\cfrac{1}{1818}\)
Теперь новый раздел, давайте анализировать
Считаем, что дробь всегда можно разложить на вид \(\cfrac{n}{p}=(b+\cfrac{c}{p})\cfrac{1}{a}\)
Какие же есть зависимости между переменными? Для этого нам надо использовать понятие Сравнения 1 степени. Теория чисел.
Бездоказательно, утверждаю что
\((b*p\:)mod \:n = -c\)
\((a*n\:)mod \:p = c\)
Можно конечно расписать, как они получились, но не в этом материале.
Что же дают эти формулы?
Ну например то, что если число \(p\) составное, и имеет делители \(d_1,d_2,d_3....d_n\)
То мы можем попытаться найти такое \(d_i\) что бы \(({p\:} mod {n})=(-1)*d_i\: mod{\:n}\)
И если такое число нашлось, то у нас получается вот так
\(\cfrac{n}{p}=(1+\cfrac{d_i}{p})\cfrac{1}{a}\)
После сокращения получаем сумму двух египетских дробей
Рассмотрим дробь \(\cfrac{13}{78125}\)
число 78125 имеет следующие делители 1, 5, 25, 125, 625, 3125, 15625, 78125
Остаток от деления 13 равен 8.
Среди делителей есть такое же число что дает остаток 5 (так как 13-8=5)?
Да есть это число 3125 и это будет параметром \(c\)
Тогда узнаем оставшийся параметр \((a*13)mod 78125 = 3125\)
\(a=6250\)
и следовательно наше разложение вот такое
\(\cfrac{13}{78125}=(1+\cfrac{3125}{78125})\cfrac{1}{6250}=\cfrac{1}{6250}+\cfrac{1}{25*6250}\)
Закрепим материал. Новая дробь \(\cfrac{41}{27368747340080916343}\)
Все алгоритмы, которые реализованы на сайтах сети Интернет, как минимум дают больше 3-х слагаемых.
Попробуем обойтись всего двумя
Остаток от деления 27368747340080916343 на 41 равно 26.
Делители числа 27368747340080916343 есть числа кратные 7-ми так \(7^23=27368747340080916343\)
Кто при делении на 41 имеет остаток 15-ть? (41-26)
Конечно же это число 343
Тогда узнаем оставшийся параметр \((a*41)mod 27368747340080916343 = 343\)
\(a=667530422928802846\) (Свои калькуляторы с такими большими числами не справились, пришлось прибегнуть к https://www.wolframalpha.com/)
и тогда наша дробь опять превращается в сумму двух слагаемых
\(\cfrac{41}{27368747340080916343}=\cfrac{1}{667530422928802846}+\cfrac{1}{79792266297612001*667530422928802846}\)
с помощью вышеуказанного ресурса и проверим истинность.
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)
Вообще, тема, где знаменатель есть число в какой либо степени - очень интересна.
Чем больше степень знаменателя - тем выше вероятность того что эту дробь можно представить в виде двух аликвотных дробей.
Обязательно расскажу об этом во второй части.
А сейчас вернемся к алгоритму. Голомб в начале 20 века описал алгоритм который в рамках выше написанного в этой статье выглядит вот так:
На каждом этапе дробь делим на сумму двух дробей \(\cfrac{n}{p}=(b+\cfrac{c}{p})\cfrac{1}{a}\)
где \(b=1\)
Тогда наши формулы
\((b*p\:)mod \:n = -c\)
\((a*n\:)mod \:p = c\)
Превращаются в
\((p\:)mod \:n = -c\)
\((a*n\:)mod \:p = c\)
Узнаем сначала \( c\) потом \( a \)
Получаем, одну аликвотную дробь и дробь вида \(\cfrac{c}{ap}\)
Повторяем все снова, пока не придем к дроби где числитель равен 1 и тогда дробь построена.
Система всегда сходится, но дробь не всегда короткая.
Но алгоритм иногда лучше чем классический алгоритм Фибоначчи.
Наш калькулятор, показывает каждый шаг разбиения, что позволит в ручном режиме найти следующий оптимальный шаг и пойти своим путем.
\(\cfrac{4}{2141}=\cfrac{1}{536}+\cfrac{1}{382526}+\cfrac{1}{1147576*191263}\)
\(\cfrac{4}{2777}=\cfrac{1}{695}+\cfrac{1}{643340}+\cfrac{1}{386003*643340}\)