O wzorach służących do obliczenia liczby liczb pierwszych nie przekraczających danej granicy

>>> Dane tekstu >>>
Autor	Antoni Baranowski
Tytuł	O wzorach służących do obliczenia liczby liczb pierwszych nie przekraczających danej granicy
Data wyd.	1895
Źródło	skany na Commons
Inne	Pobierz jako: EPUB  • PDF  • MOBI
Indeks stron

[3]

Ks. Biskup A. Baranowski.

---

O WZORACH

SŁUŻĄCYCH DO

OBLICZENIA LICZBY LICZB PIERWSZYCH

NIE PRZEKRACZAJĄCYCH DANEJ GRANICY.

W KRAKOWIE.

NAKŁADEM AKADEMII UMIEJĘTNOŚCI.

SKŁAD GŁÓWNY W KSIĘGARNI SPÓŁKI WYDAWNICZEJ POLSKIEJ.

1895.

[4]

Osobne odbicie z Tomu XXVIII. Rozpraw Wydziału matematyczno-przyrodniczego
Akademii Umiejętności w Krakowie.

Kraków, 1895. — W drukarni Uniwersytetu Jagiellońskiego, pod zarządem A. M. Kosterkiewicza.

[5]

O wzorach

służących do obliczenia liczby liczb pierwszych

nie przekraczających danej granicy.

Przez

ks. biskupa A. Baranowskiego.

---

Rzecz przedstawiona na posiedzeniu Wydz. mat.-przyr. z dnia 2. lipca 1894 r.;

referent czł. Mertens.

 Od lat kilku zupełnie zależałem pole gramatyki, a ugrzązłem, raczej pogrążyłem się w matematyce, nie jako fachowy, ale jak dyletant, samouczek, bez prawideł, wskazówek i znajomości powszechnie przyjętych formuł. Napisałem kwadraty i sześciany liczb od $1$ do $1000$ z ich różnicami. Potęgowałem liczby małe: $2,$ $2^{2},$ $2^{3}\ldots$ $2^{300};$ $3,$ $3^{2},$ $3^{3}\ldots$ $3^{200};$ $5,$ $5^{2},$ $5^{3}\ldots$ $5^{150};$ $7,$ $7^{2},$ $7^{3}\ldots$ $7^{85}.$ Przy tej pracy poznałem stosunki i prawidła, o których później się dowiedziałem, że nazywane są dwumianem Newton’a. Zastanawiając się nad stosunkami i prawami wykładników, wpadłem na trop, który mię doprowadził do zbadania wielokształtu podzielności liczb złożonych. Przytem zbadałem wszystkie liczby w zakresie numeracyi od $0$ do $1000$; potem do $10\,000$. Za wskazówką i poradą Dra Webera zapisałem sobie dzieło Wertheima „Die Theorie der Zahlen“, ale jego wykład i znakowanie — dla mnie, jako laika, były orzechami zanadto twardemi na moje zęby. Zainteresowała mię jednak funkcya $\operatorname {\varphi } (m)$, jako mająca związek z moją analizą liczb w celu znalezienia prawideł, zapomocą których mógłbym poznać, która liczba jest pierwszą, [6]a jakie są cechy liczb złożonych. Przy pomocy Dra Hossfelda w Eisenach, poznałem zasady funkcyi $\operatorname {\varphi } (m)$. Opracowując tę funkcyę w zakresie do $100\,000$, potem do $1,000\,000$, wykryłem okresy symetrycznych luk między liczbami względnie pierwszemi, po usunięciu każdej z porządku z liczb pierwszych ze wszystkiemi przez nią podzielnemi liczbami i prawidło ogólne, że

	$\operatorname {\varphi } (p_{1}.p_{2}.p_{3}\ldots p_{n},n)=(p_{1}-1)(p_{2}-1)(p_{3}-1)\ldots (p_{n}-1)$
oraz że	$\operatorname {\varphi } (g(p_{1}.p_{2}.p_{3}\ldots p_{n},n))=g(p_{1}-1)(p_{2}-1)(p_{3}-1)\ldots (p_{n}-1)$.

 Wykryta zaś symetrya luk w danym okresie, t. j. że luka 1-sza $=$ luce ostatniej, 2-ga $=$ przedostatniej , 3-cia z początku $=$ 3-ciej od końca i t. d. aż do połowy, czyli środka okresu, to jest do ${{p_{1}.p_{2}.p_{3}\ldots p_{n}} \over 2}$ wskazała możność obliczenia na poczekaniu i zakresu numeracyi
$0,1,2,3,\ldots m=p_{1}.p_{2}.p_{3}\ldots p_{n}+r$,
oraz $m=g.p_{1}.p_{2}.p_{3}\ldots p_{n}\pm {r}$,
gdyż $\operatorname {\varphi } (p_{1}.p_{2}.p_{3}\ldots p_{n}\pm {r},n)=(p_{1}-1)(p_{2}-1)(p_{3}-1)\ldots (p_{n}-1){\begin{cases}+\operatorname {\varphi } (r,n)\\-\operatorname {\varphi } (r-1,n),\end{cases}}$
oraz $\operatorname {\varphi } (g.(p_{1}.p_{2}.p_{3}\ldots p_{n}\pm {r},n))=(g(p_{1}-1)(p_{2}-1)(p_{3}-1)\ldots (p_{n}-1){\begin{cases}+\operatorname {\varphi } (r,n)\\-\operatorname {\varphi } (r-1,n).\end{cases}}$
 Zakomunikowałem te spostrzeżenia przez p. Webera Drowi Hossfeldowi, który przyznał im wielką wagę; ale potem zawiadomił mię, że już w roku 1872 Dr. Meissel te same prawa odkrył i ogłosił, tylko jeszcze nie wiedział o wewnętrznej symetryi okresów i dla tego wzór jego posiada tylko $+r,$ ale nie $-r.$ Wydrukował o tem za mojem przyzwoleniem króciutki artykulik o dopełnieniu przeze mnie wzoru Meissela. Artykulik wyszedł niefortunnie, bo przed wydrukowaniem nie przysłał mi go do przejrzenia. Wsadził doń bez potrzeby własnego pomysłu $p_{x},$ bez potrzeby i sensu. Kiedym go później przekonał, iż wszystkie trudności i pomyłki wyrażenie $-\operatorname {\varphi } (r-1,n)$ usuwa, przysłał mi do przejrzenia napisany do druku drugi artykulik o tem samem; ale jego objaśnienie $-\operatorname {\varphi } (r-1,n),-1$, nie trafiło do mojego przekonania; moje zaś objaśnienie on uważał za naciągnięte. W ten sposób artykulik jego nie wyszedł z druku.
 Dla większego jeszcze uproszczenia obliczeń funkcyi $\operatorname {\varphi } (m)$, ułożyłem analityczną tablicę okresu $p_{6},$ t. j. $\operatorname {\varphi } (2.3.5.7.11.13,6)=\operatorname {\varphi } (30030,6)$, która wielkie daje ułatwienie przy obliczaniu większych zakresów numeracyi.
[7] Potem napisałem teoryę luk, których Dr. Hossfeld nie podjął się sprawdzić.
 Nareszcie usunąwszy wszystkie liczby podzielne przez $2,$ $3$ i $5$, jako łatwe do poznania, zanalizowałem zakres numeracyi $0,$ $1,$ $2,$ $3,\ldots$ $150\,060$, czyli liczby wszystkie pierwsze w tym zakresie, oraz podzielne przez $7,$ $11,$ $13,$ $17,\ldots ,$ t. j. przez $p_{4},$ $p_{5},$ $p_{6},$ $p_{7},\ldots$ Spisałem na ogół liczb $40\,008$, oznaczając je właściwemi czynnikami n. p. $49=7^{2},$ $77=7.11;$ $91=7.13,$ $1001=7.11.13$ i t. d.
 Później z tego kajetu wypisałem osobno same tylko liczby pierwsze, to jest:

$p_{1}=2$, $p_{2}=3$, $p_{3}=5$, $p_{4}=7$ .... $p_{13852}=150\,053$.

 Gustaw Wertheim w dziele „Elemente der Zahlentheorie“ (Leipzig 1887) rozwija i przykładem objaśnia następujący wzór Meissel’a do obliczenia w danym zakresie numeracyi liczb pierwszych str. 24.

$\operatorname {\psi } (n)=\operatorname {\varphi } (n,m)+m(\mu +1)+{{\mu (\mu -1)} \over 2}-1-\sum _{s=1}^{s=\mu }\operatorname {\psi } {\biggl (}{n \over {p_{m+s}}}{\biggr )}$

$\operatorname {\psi } (n)$ oznacza tutaj, ile się zawiera liczb bezwzględnie pierwszych w zakresie numeracyi od $0,$ $1,$ $2,$ $3,\ldots$ $n.$
 $m$ oznacza, ile liczb pierwszych znajduje się w sześciennym pierwiastku zakresu $n$, czyli $m=\operatorname {\varphi } {\Bigl (}{\sqrt[{3}]{n}}{\Bigr )}$.
 $\mu$ oznacza, ile liczb pierwszych znajduje się w pierwiastku kwadratowym ${\sqrt {2}}$ , po odjęciu liczby tychże liczb, będących w pierwiastku sześciennym, czyli $\mu =\psi {\sqrt {n}}-m$.

$m+\mu =\psi {\Bigl (}{\sqrt {n}}{\Bigr )}$.

 Wzór ten, dobry przy obliczaniu niewielkich zakresów numeracyi, kiedy $n$ nie przewyższa setek, tysięcy; znośny jeszcze i przy obliczaniu dziesiątek tysięcy; w wielkich zaś zakresach numeracyi, wymaga wiele miejsca, czasu i pracy. Można się o tem przekonać, obliczając choćby tylko

$n=100\,000$; ${\sqrt[{3}]{n}}=46$; $\operatorname {\psi } (46)=14$; ${\sqrt {n}}=316$; $\operatorname {\psi } (316)=65$;

ponieważ zaś $m=\operatorname {\psi } ({\sqrt[{3}]{n}})=\operatorname {\psi } (46)=14,$ [8]przeto $\mu =\left[\psi \left({\sqrt {n}}\right)=\psi \left(316\right)=65\right]-14=51;$ $m+\mu =\psi \left({\sqrt {n}}\right)=65$
$\varphi (n,m)=\varphi (100\,000,14)=14204$. Tego trzeba dowieść rachunkiem następującym.
 Liczby te 14 są następujące: $p_{1}=2,$ $p_{2}=3,$ $p_{3}=5,$ $p_{4}=7,$ $p_{5}=11,$ $p_{6}=13,$ $p_{7}=17,$ $p_{8}=19,$ $p_{9}=23,$ $p_{10}=29,$ $p_{11}=31,$ $p_{12}=37,$ $p_{13}=41,$ $p_{14}=43;$

$\varphi (100\,000,14)=\varphi (100\,000,13)-\left[\varphi \left({\frac {100\,000}{43}},13\right)=\varphi \left(\,\,\,\,2325,13\right)\right]=14\,540-\,\,\,\,\,\,\,336=14\,204\,\,$

$\varphi (100\,000,13)=\varphi (100\,000,12)-\left[\varphi \left({\frac {100\,000}{41}},12\right)=\varphi \left(\,\,\,\,2439,12\right)\right]=14\,899-\,\,\,\,\,\,\,359=14\,540\,\,$

$\varphi (100\,000,12)=\varphi (100\,000,11)-\left[\varphi \left({\frac {100\,000}{37}},11\right)=\varphi \left(\,\,\,\,2707,11\right)\right]=15\,305-\,\,\,\,\,\,\,406=14\,899\,\,$

$\varphi (100\,000,11)=\varphi (100\,000,10)-\left[\varphi \left({\frac {100\,000}{31}},10\right)=\varphi \left(\,\,\,\,3225,10\right)\right]=15\,805-\,\,\,\,\,\,\,500=15\,305\,\,$

$\varphi (100\,000,10)=\varphi (100\,000,\,\,\,9)-\left[\varphi \left({\frac {100\,000}{29}},\,\,\,9\right)=\varphi \left(\,\,\,\,3448,\,\,\,9\right)\right]=16\,361-\,\,\,\,\,\,\,556=15\,805\,\,$

$\varphi (100\,000,\,\,\,9)=\varphi (100\,000,\,\,\,8)-\left[\varphi \left({\frac {100\,000}{23}},\,\,\,8\right)=\varphi \left(\,\,\,\,4347,\,\,\,8\right)\right]=17\,103-\,\,\,\,\,\,\,742=16\,361\,\,$

$\varphi (100\,000,\,\,\,8)=\varphi (100\,000,\,\,\,7)-\left[\varphi \left({\frac {100\,000}{19}},\,\,\,7\right)=\varphi \left(\,\,\,\,5263,\,\,\,7\right)\right]=18\,053-\,\,\,\,\,\,\,950=17\,103\,\,$

$\varphi (100\,000,\,\,\,7)=\varphi (100\,000,\,\,\,6)-\left[\varphi \left({\frac {100\,000}{17}},\,\,\,6\right)=\varphi \left(\,\,\,\,5882,\,\,\,6\right)\right]=19\,181-\,\,\,\,1128=18\,053\,\,$

$\varphi (100\,000,\,\,\,6)=\varphi (100\,000,\,\,\,5)-\left[\varphi \left({\frac {100\,000}{13}},\,\,\,5\right)=\varphi \left(\,\,\,\,7692,\,\,\,5\right)\right]=20\,779-\,\,\,\,1598=19\,181\,\,$

$\varphi (100\,000,\,\,\,5)=\varphi (100\,000,\,\,\,4)-\left[\varphi \left({\frac {100\,000}{11}},\,\,\,4\right)=\varphi \left(\,\,\,\,9090,\,\,\,4\right)\right]=22\,857-\,\,\,\,2078=20\,779\,\,$

[9]

$\varphi (100\,000,\,\,\,4)=\varphi (100\,000,\,\,\,3)-\left[\varphi \left({\frac {100\,000}{7}},\,\,\,3\right)=\varphi \left(14\,285,\,\,\,3\right)\right]=26\,666-\,\,\,\,3809=22\,857\,\,$

$\varphi (100\,000,\,\,\,3)=\varphi (100\,000,\,\,\,2)-\left[\varphi \left({\frac {100\,000}{5}},\,\,\,2\right)=\varphi \left(20\,000,\,\,\,2\right)\right]=33\,333-\,\,\,\,6667=26\,666\,\,$

$\varphi (100\,000,\,\,\,2)=\varphi (100\,000,\,\,\,1)-\left[\varphi \left({\frac {100\,000}{3}},\,\,\,1\right)=\varphi \left(33\,333,\,\,\,1\right)\right]=50\,000-16\,667=33\,333.$

[10] [11] [12] [13] [14]

$\varphi (20\,000,2)=\varphi (20\,000,1)-\varphi (6\,666,1)=10\,000-3\,333$	$=$	$6\,667$
$\varphi (14\,285,3)=\varphi (14\,285,2)-\varphi (2\,857,2)=4\,762-953=3\,809$ $\varphi (14\,285,2)=\varphi (14\,285,1)-\varphi (4\,761,1)=7\,143-2\,381=4\,762$ $\varphi (2857,2)=\varphi (2\,857,1)-\varphi (952,1)=1\,429-476$	$=$	$953$
$\varphi (9\,090,4)=\varphi (9\,090,3)-\varphi (1\,298,3)=2\,424-346=2\,078$ $\varphi (9\,090,3)=\varphi (9\,090,2)-\varphi (1\,818,2)=3\,030-606=2\,424$ $\varphi (9\,090,2)=\varphi (9\,090,1)-\varphi (3\,030,1)=4\,545-1\,515=3\,030$ $\varphi (1\,818,2)=\varphi (1\,818,1)-\varphi (606,1)=909-303$	$=$	$606$
$\varphi (1\,298,3)=\varphi (1\,298,2)-\varphi (259,2)=433-87=346$ $\varphi (1\,298,2)=\varphi (1\,298,1)-\varphi (432,1)=649-216=433$ $\varphi (259,2)=\varphi (259,1)-\varphi (86,1)=130-43$	$=$	$87$
$\varphi (7\,692,5)=\varphi (7\,692,4)-\varphi (699,4)=1\,758-160=1\,598$ $\varphi (7\,692,4)=\varphi (7\,692,3)-\varphi (1\,098,3)=2\,051-293=1\,758$ $\varphi (7\,692,3)=\varphi (7\,692,2)-\varphi (1\,538,2)=2\,564-513=2\,051$ $\varphi (7\,692,2)=\varphi (7\,692,1)-\varphi (2\,564,1)=3\,846-1\,282=2\,564$ $\varphi (1\,538,2)=\varphi (1\,538,1)-\varphi (512,1)=769-256$	$=$	$513$
$\varphi (1\,098,3)=\varphi (1\,098,2)-\varphi (219,2)=366-73=293$ $\varphi (1\,098,2)=\varphi (1\,098,1)-\varphi (366,1)=549-183=366$ $\varphi (219,2)=\varphi (219,1)-\varphi (73,1)=110-37$	$=$	$73$
$\varphi (699,4)=\varphi (699,3)-\varphi (99,3)=186-26=160$ $\varphi (699,3)=\varphi (699,2)-\varphi (139,2)=233-47=186$ $\varphi (699,2)=\varphi (699,1)-\varphi (233,1)=350-117=233$ $\varphi (139,2)=\varphi (139,1)-\varphi (46,1)=70-23$	$=$	$47$
$\varphi (99,3)=\varphi (99,2)-\varphi (19,2)=33-7=26$ $\varphi (99,2)=\varphi (99,1)-\varphi (33,1)=50-17=33$ $\varphi (19,2)=\varphi (19,1)-\varphi (6,1)=10-3$	$=$	$7$
$\varphi (5882,6)=\varphi (5882,5)-\varphi (452,5)=1222-94=1128$ $\varphi (5882,5)=\varphi (5882,4)-\varphi (534,4)=1345-123=1222$ $\varphi (5882,4)=\varphi (5882,3)-\varphi (840,3)=1569-224=1345$ $\varphi (5882,3)=\varphi (5882,2)-\varphi (1176,2)=1961-392=1569$ $\varphi (5882,2)=\varphi (5882,1)-\varphi (1960,1)=2941-980=1961$ $\varphi (1176,2)=\varphi (1176,1)-\varphi (391,1)=588-196$	$=$	$392$
$\varphi (840,3)=\varphi (840,2)-\varphi (168,2)=280-56=224$ $\varphi (840,2)=\varphi (840,1)-\varphi (280,1)=420-140=280$ $\varphi (168,2)=\varphi (168,1)-\varphi (56,1)=84-28$	$=$	$56$
$\varphi (534,4)=\varphi (534,3)-\varphi (76,3)=143-20=123$ $\varphi (534,3)=\varphi (534,2)-\varphi (106,2)=178-35=143$ $\varphi (534,2)=\varphi (534,1)-\varphi (178,1)=267-89=178$ $\varphi (106,2)=\varphi (106,1)-\varphi (35,1)=53-18$	$=$	$35$
$\varphi (76,3)=\varphi (76,2)-\varphi (15,2)=25-5=20$ $\varphi (76,2)=\varphi (76,1)-\varphi (25,1)=38-13=25$ $\varphi (15,2)=\varphi (15,1)-\varphi (5,1)=8-3$	$=$	$5$
$\varphi (452,5)=\varphi (452,4)-\varphi (41,4)=104-10=94$ $\varphi (452,4)=\varphi (452,3)-\varphi (64,3)=121-17=104$ $\varphi (452,3)=\varphi (452,2)-\varphi (90,2)=151-30=121$ $\varphi (452,2)=\varphi (452,1)-\varphi (150,1)=226-75=151$ $\varphi (90,2)=\varphi (90,1)-\varphi (90,1)=45-15$	$=$	$30$
$\varphi (64,3)=\varphi (64,2)-\varphi (12,2)=21-4=17$ $\varphi (64,2)=\varphi (64,1)-\varphi (12,1)=32-11=21$ $\varphi (12,2)=\varphi (12,1)-\varphi (4,1)=6-2$	$=$	$4$
$\varphi (41,4)=\varphi (41,3)-\varphi (5,3)=11-1=10$ $\varphi (41,3)=\varphi (41,2)-\varphi (8,2)=14-3=11$ $\varphi (41,2)=\varphi (41,1)-\varphi (13,1)=21-7=14$
$\varphi (5263,7)=\varphi (5263,6)-\varphi (309,6)=1009-59=950$ $\varphi (5263,6)=\varphi (5263,5)-\varphi (404,5)=1094-85=1009$ $\varphi (5263,5)=\varphi (5263,4)-\varphi (478,4)=1203-109=1094$ $\varphi (5263,4)=\varphi (5263,3)-\varphi (751,3)=1404-201=1203$ $\varphi (5263,3)=\varphi (5263,2)-\varphi (1052,2)=1755-351=1404$ $\varphi (5263,2)=\varphi (5263,1)-\varphi (1754,1)=2632-877=1755$ $\varphi (1052,2)=\varphi (1052,1)-\varphi (350,1)=526-175$	$=$	$351$
$\varphi (751,3)=\varphi (751,2)-\varphi (150,2)=251-\,\,\,50=201$ $\varphi (751,2)=\varphi (751,1)-\varphi (250,1)=376-125=251$ $\varphi (150,2)=\varphi (150,1)-\varphi (50,1)=75-25$	$=$	$50$
$\varphi (478,4)=\varphi (478,3)-\varphi (68,3)=127-18=109$ $\varphi (478,3)=\varphi (478,2)-\varphi (95,2)=159-32=127$ $\varphi (478,2)=\varphi (478,1)-\varphi (159,1)=239-80=159$ $\varphi (95,2)=\varphi (95,1)-\varphi (31,1)=48-16$	$=$	$32$
$\varphi (68,3)=\varphi (68,2)-\varphi (13,2)=23-5=18$ $\varphi (68,2)=\varphi (68,1)-\varphi (22,1)=34-11=23$ $\varphi (13,2)=\varphi (13,1)-\varphi (4,1)=7-2$	$=$	$5$
$\varphi (404,5)=\varphi (404,4)-\varphi (36,4)=93-8=85$ $\varphi (404,4)=\varphi (404,3)-\varphi (57,3)=108-15=93$ $\varphi (404,3)=\varphi (404,2)-\varphi (80,2)=135-27=108$ $\varphi (404,2)=\varphi (404,1)-\varphi (134,1)=202-67=135$ $\varphi (80,2)=\varphi (80,1)-\varphi (26,1)=40-13$	$=$	$27$
$\varphi (57,3)=\varphi (57,2)-\varphi (11,2)=19-4=15$ $\varphi (57,2)=\varphi (57,1)-\varphi (19,1)=29-10=19$ $\varphi (11,2)=\varphi (11,1)-\varphi (3,1)=6-2$	$=$	$4$
$\varphi (36,4)=\varphi (36,3)-\varphi (5,3)=9-1=8$ $\varphi (36,3)=\varphi (36,2)-\varphi (7,2)=12-3=9$ $\varphi (36,2)=\varphi (36,1)-\varphi (12,1)=18-6=12$ $\varphi (7,2)=\varphi (7,1)-\varphi (2,1)=4-1$	$=$	$3$
$\varphi (5,3)=\varphi (5,2)-\varphi (1,2)=2-1=1$ $\varphi (5,2)=\varphi (5,1)-\varphi (1,1)=3-1=2$
$\varphi (309,6)=\varphi (309,5)-\varphi (23,5)=64-5=59$ $\varphi (309,5)=\varphi (309,4)-\varphi (28,4)=70-6=64$ $\varphi (309,4)=\varphi (309,3)-\varphi (44,3)=82-12=70$ $\varphi (309,3)=\varphi (309,2)-\varphi (61,2)=103-21=82$ $\varphi (309,2)=\varphi (309,1)-\varphi (103,1)=155-52=103$ $\varphi (61,2)=\varphi (61,1)-\varphi (20,1)=31-10$	$=$	$21$
$\varphi (44,3)=\varphi (44,2)-\varphi (8,2)=15-3=12$ $\varphi (44,2)=\varphi (44,1)-\varphi (14,1)=22-7=15$ $\varphi (8,2)=\varphi (8,1)-\varphi (2,1)=4-1$	$=$	$3$
$\varphi (25,4)=\varphi (28,3)-\varphi (4,3)=7-1=6$ $\varphi (28,3)=\varphi (28,2)-\varphi (5,2)=9-2=7$ $\varphi (28,2)=\varphi (28,1)-\varphi (9,1)=14-5=9$ $\varphi (5,2)=\varphi (5,1)-\varphi (1,1)=3-1$	$=$	$2$
$\varphi (23,5)=\varphi (23,4)-\varphi (2,4)=\,\,\,6-1=5$ $\varphi (23,4)=\varphi (23,3)-\varphi (3,3)=\,\,\,7-1=6$ $\varphi (23,3)=\varphi (23,2)-\varphi (4,2)=\,\,\,8-1=7$ $\varphi (23,2)=\varphi (23,1)-\varphi (7,1)=12-4=8$	${\displaystyle }$	${\displaystyle }$
$\varphi (4347,8)=\varphi (4347,7)-\varphi (\,\,\,228,7)=\,\,\,785-\,\,\,43=\,\,\,742$ $\varphi (4347,7)=\varphi (4347,6)-\varphi (\,\,\,255,6)=\,\,\,834-\,\,\,49=\,\,\,785$ $\varphi (4347,6)=\varphi (4347,5)-\varphi (\,\,\,334,5)=\,\,\,903-\,\,\,69=\,\,\,834$ $\varphi (4347,5)=\varphi (4347,4)-\varphi (\,\,\,395,4)=\,\,\,993-\,\,\,90=\,\,\,903$ $\varphi (4347,4)=\varphi (4347,3)-\varphi (\,\,\,621,3)=1159-166=\,\,\,993$ $\varphi (4347,3)=\varphi (4347,2)-\varphi (\,\,\,869,2)=1149-290=1159$ $\varphi (4347,2)=\varphi (4347,1)-\varphi (1449,1)=2174-725=1449$ $\varphi (869,2)=\varphi (869,1)-\varphi (289,1)=435-\,\,\,145$	$=$	$290$
$\varphi (621,3)=\varphi (621,2)-\varphi (124,2)=207-\,\,\,41$	$=$	$164$
$\varphi (621,2)=\varphi (621,1)-\varphi (207,1)=311-104$	$=$	$207$
$\varphi (124,2)=\varphi (124,1)-\varphi (41,1)=\,\,\,62-\,\,\,21$	$=$	$41$
$\varphi (395,4)=\varphi (395,3)-\varphi (56,3)=105-15=\,\,\,90$ $\varphi (395,3)=\varphi (395,2)-\varphi (79,2)=132-27=105$ $\varphi (395,2)=\varphi (395,1)-\varphi (131,1)=198-66=132$ $\varphi (79,2)=\varphi (79,1)-\varphi (26,1)=40-13$	$=$	$27$
$\varphi (56,3)=\varphi (56,2)-\varphi (11,2)=19-4=15$ $\varphi (56,2)=\varphi (56,1)-\varphi (18,1)=28-9=19$ $\varphi (11,2)=\varphi (11,1)-\varphi (3,1)=6-\,\,\,2$	$=$	$4$
$\varphi (334,5)=\varphi (334,4)-\varphi (\,\,\,30,4)=\,\,\,76-\,\,\,7=\,\,\,69$ $\varphi (334,4)=\varphi (334,3)-\varphi (\,\,\,47,3)=\,\,\,89-13=\,\,\,76$ $\varphi (334,3)=\varphi (334,2)-\varphi (\,\,\,66,2)=111-22=\,\,\,89$ $\varphi (334,2)=\varphi (334,1)-\varphi (111,1)=167-56=111$ $\varphi (66,2)=\varphi (66,1)-\varphi (22,1)=33-\,\,\,11$	$=$	$22$
$\varphi (47,3)=\varphi (47,2)-\varphi (\,\,\,9,2)=\,\,\,\,\,\,\,\,\,\,\,16-\,\,\,\,\,\,\,\,\,\,\,3$	$=$	$13$
$\varphi (47,2)=\varphi (47,1)-\varphi (15,1)=\,\,\,\,\,\,\,\,\,\,\,24-\,\,\,\,\,\,\,\,\,\,\,8$	$=$	$16$
$\varphi (\,\,\,9,2)=\varphi (\,\,\,9,1)-\varphi (3,1)=5-2$	$=$	$\,\,\,3$
$\varphi (30,4)=\varphi (30,3)-\varphi (\,\,\,4,3)=\,\,\,8-1=\,\,\,7$ $\varphi (30,3)=\varphi (30,2)-\varphi (\,\,\,6,2)=10-2=\,\,\,8$ $\varphi (30,2)=\varphi (30,1)-\varphi (10,1)=15-5=10$	${\displaystyle }$	${\displaystyle }$
$\varphi (255,6)=\varphi (255,5)-\varphi (19,5)=\,\,\,\,\,\,53\,\,\,\,\,\,\,-\,\,\,4=49$  $\varphi (255,5)=\varphi (255,4)-\varphi (23,4)=\,\,\,\,\,\,59\,\,\,\,\,\,\,-\,\,\,6=53$  $\varphi (255,4)=\varphi (255,3)-\varphi (36,3)=\,\,\,\,\,\,68\,\,\,\,\,\,\,-\,\,\,9=59$  $\varphi (255,3)=\varphi (255,2)-\varphi (51,2)=\,\,\,\,\,\,85\,\,\,\,\,\,\,-17=68$  $\varphi (255,2)=\varphi (255,1)-\varphi (85,1)=\,\,\,128\,\,\,\,\,\,\,-43=85$  $\varphi (\,\,\,51,2)=\varphi (51,1)-\varphi (17,1)=26-\,\,\,9$	$=$	$17$
$\varphi (36,3)=\varphi (36,2)-\varphi (\,\,\,7,2)=\,\,\,\,\,\,\,\,\,\,\,12-3=\,\,\,9$ $\varphi (36,2)=\varphi (36,1)-\varphi (12,1)=\,\,\,\,\,\,\,\,\,\,\,18-6=12$ $\varphi (\,\,\,7,2)=\varphi (\,\,\,7,1)-\varphi (2,1)=4-\,\,\,1$	$=$	$3$
$\varphi (23,4)=\varphi (23,3)-\varphi (3,3)=\,\,\,7-1=6$ $\varphi (23,3)=\varphi (23,2)-\varphi (4,2)=\,\,\,8-1=7$ $\varphi (23,2)=\varphi (23,1)-\varphi (7,1)=12-4=8$
$\varphi (19,5)=\varphi (19,4)-\varphi (1,4)=\,\,\,5-1=4$ $\varphi (19,4)=\varphi (19,3)-\varphi (2,3)=\,\,\,6-1=5$ $\varphi (19,3)=\varphi (19,2)-\varphi (3,2)=\,\,\,7-1=6$ $\varphi (19,2)=\varphi (19,1)-\varphi (6,1)=10-3=7$
$\varphi (228,7)=\varphi (228,6)-\varphi (13,6)=\,\,\,\,\,\,44\,\,\,\,\,\,\,\,-\,\,\,1=43$  $\varphi (228,6)=\varphi (228,5)-\varphi (17,5)=\,\,\,\,\,\,47\,\,\,\,\,\,\,\,-\,\,\,3=44$  $\varphi (228,5)=\varphi (228,4)-\varphi (20,4)=\,\,\,\,\,\,52\,\,\,\,\,\,\,\,-\,\,\,5=47$  $\varphi (228,4)=\varphi (228,3)-\varphi (32,3)=\,\,\,\,\,\,61\,\,\,\,\,\,\,\,-\,\,\,9=52$  $\varphi (228,3)=\varphi (228,2)-\varphi (45,2)=\,\,\,\,\,\,76\,\,\,\,\,\,\,\,-15=61$  $\varphi (228,2)=\varphi (228,1)-\varphi (76,1)=\,\,\,114\,\,\,\,\,\,\,\,-38=76$  $\varphi (\,\,\,45,2)=\varphi (45,1)-\varphi (15,1)=23-\,\,\,8$	$=$	$15$
$\varphi (32,3)=\varphi (32,2)-\varphi (\,\,\,6,2)=11-2$	$=$	$9$
$\varphi (32,2)=\varphi (32,1)-\varphi (10,1)=16-5$	$=$	$11$
$\varphi (3448,9)=\varphi (3448,8)-\varphi (\,\,\,149,8)=\,\,\,\,\,\,584\,\,\,\,\,\,\,\,-\,\,\,28=\,\,\,556$ $\varphi (3448,8)=\varphi (3448,7)-\varphi (\,\,\,181,7)=\,\,\,\,\,\,620\,\,\,\,\,\,\,\,-\,\,\,36=\,\,\,584$ $\varphi (3448,7)=\varphi (3448,6)-\varphi (\,\,\,202,6)=\,\,\,\,\,\,661\,\,\,\,\,\,\,\,-\,\,\,41=\,\,\,620$ $\varphi (3448,6)=\varphi (3448,5)-\varphi (\,\,\,265,5)=\,\,\,\,\,\,716\,\,\,\,\,\,\,\,-\,\,\,55=\,\,\,661$ $\varphi (3448,5)=\varphi (3448,4)-\varphi (\,\,\,313,4)=\,\,\,\,\,\,788\,\,\,\,\,\,\,\,-\,\,\,72=\,\,\,716$ $\varphi (3448,4)=\varphi (3448,3)-\varphi (\,\,\,492,3)=\,\,\,\,\,\,919\,\,\,\,\,\,\,\,-131=\,\,\,788$ $\varphi (3448,3)=\varphi (3448,2)-\varphi (\,\,\,689,2)=\,\,\,1149\,\,\,\,\,\,\,\,-230=\,\,\,919$ $\varphi (3448,2)=\varphi (3448,1)-\varphi (1149,1)=\,\,\,1724\,\,\,\,\,\,\,\,-575=1149$ $\varphi (\,\,\,689,2)=\varphi (\,\,\,689,1)-\varphi (229,1)=345-\,\,\,115$	$=$	$230$
$\varphi (492,3)=\varphi (492,2)-\varphi (\,\,\,98,2)=164-33$	$=$	$131$
$\varphi (492,2)=\varphi (491,1)-\varphi (164,1)=246-82$	$=$	$164$
$\varphi (98,2)=\varphi (98,1)-\varphi (32,1)=49-16$	$=$	$33$

$\varphi (313,4)=\varphi (313,3)-\varphi (\,\,\,44,3)=\,\,\,\,\,\,84\,\,\,\,\,\,\,-12=\,\,\,72$  $\varphi (313,3)=\varphi (313,2)-\varphi (\,\,\,62,2)=\,\,\,105\,\,\,\,\,\,\,-21=\,\,\,84$  $\varphi (313,2)=\varphi (313,1)-\varphi (104,1)=\,\,\,157\,\,\,\,\,\,\,-52=105$  $\varphi (\,\,\,62,2)=\varphi (62,1)-\varphi (20,1)=31-\,\,\,10$	$=$	$21$
$\varphi (\,\,\,44,3)=\varphi (44,2)-\varphi (\,\,\,8,2)=15-\,\,\,\,\,\,3$  $\varphi (\,\,\,44,2)=\varphi (44,1)-\varphi (12,1)=22-\,\,\,\,\,\,7$	$=$ $=$	$12$ $15$
$\varphi (265,5)=\varphi (265,4)-\varphi (\,\,\,24,4)=\,\,\,\,\,\,61\,\,\,\,\,\,\,-\,\,\,6=\,\,\,55$  $\varphi (265,4)=\varphi (265,3)-\varphi (\,\,\,37,3)=\,\,\,\,\,\,71\,\,\,\,\,\,\,-10=\,\,\,61$  $\varphi (265,3)=\varphi (265,2)-\varphi (\,\,\,53,2)=\,\,\,\,\,\,89\,\,\,\,\,\,\,-18=\,\,\,71$  $\varphi (265,2)=\varphi (265,1)-\varphi (\,\,\,88,1)=\,\,\,133\,\,\,\,\,\,\,-44=\,\,\,89$  $\varphi (\,\,\,53,2)=\varphi (53,1)-\varphi (17,1)=27-\,\,\,\,\,\,9$	$=$	$18$
$\varphi (\,\,\,37,3)=\varphi (37,2)-\varphi (\,\,\,7,2)=13-\,\,\,\,\,\,3$  $\varphi (\,\,\,37,2)=\varphi (37,1)-\varphi (14,1)=19-\,\,\,\,\,\,6$	$=$ $=$	$10$ $13$
$\varphi (202,6)=\varphi (202,5)-\varphi (\,\,\,15,5)=\,\,\,\,\,\,43\,\,\,\,\,\,\,-\,\,\,2=\,\,\,41$  $\varphi (202,5)=\varphi (202,4)-\varphi (\,\,\,18,4)=\,\,\,\,\,\,47\,\,\,\,\,\,\,-\,\,\,4=\,\,\,43$  $\varphi (202,4)=\varphi (202,3)-\varphi (\,\,\,28,3)=\,\,\,\,\,\,54\,\,\,\,\,\,\,-\,\,\,7=\,\,\,47$  $\varphi (202,3)=\varphi (202,2)-\varphi (\,\,\,40,2)=\,\,\,\,\,\,67\,\,\,\,\,\,\,-13=\,\,\,54$  $\varphi (202,2)=\varphi (202,1)-\varphi (\,\,\,67,1)=\,\,\,101\,\,\,\,\,\,\,-34=\,\,\,67$
$\varphi (181,7)=\varphi (181,6)-\varphi (\,\,\,10,6)=\,\,\,\,\,\,37\,\,\,\,\,\,\,-\,\,\,1=\,\,\,36$  $\varphi (181,6)=\varphi (181,5)-\varphi (\,\,\,13,5)=\,\,\,\,\,\,39\,\,\,\,\,\,\,-\,\,\,2=\,\,\,37$  $\varphi (181,5)=\varphi (181,4)-\varphi (\,\,\,16,4)=\,\,\,\,\,\,42\,\,\,\,\,\,\,-\,\,\,3=\,\,\,39$  $\varphi (181,4)=\varphi (181,3)-\varphi (\,\,\,25,3)=\,\,\,\,\,\,49\,\,\,\,\,\,\,-\,\,\,7=\,\,\,42$  $\varphi (181,3)=\varphi (181,2)-\varphi (\,\,\,36,2)=\,\,\,\,\,\,61\,\,\,\,\,\,\,-12=\,\,\,49$  $\varphi (181,2)=\varphi (181,1)-\varphi (\,\,\,50,1)=\,\,\,\,\,\,91\,\,\,\,\,\,\,-30=\,\,\,61$  $\varphi (\,\,\,36,2)=\varphi (36,1)-\varphi (12,1)=18-\,\,\,\,\,\,6$	$=$	$12$
$\varphi (\,\,\,25,3)=\varphi (25,2)-\varphi (\,\,\,5,2)=\,\,\,9-\,\,\,\,\,\,2$  $\varphi (\,\,\,25,2)=\varphi (25,1)-\varphi (\,\,\,8,1)=13-\,\,\,\,\,\,4$	$=$ $=$	$8$ $9$
$\varphi (149,8)=\varphi (149,7)-\varphi (\,\,\,\,\,\,7,7)=\,\,\,\,\,\,29\,\,\,\,\,\,\,-\,\,\,1=\,\,\,28$  $\varphi (149,7)=\varphi (149,6)-\varphi (\,\,\,\,\,\,8,6)=\,\,\,\,\,\,30\,\,\,\,\,\,\,-\,\,\,1=\,\,\,29$  $\varphi (149,6)=\varphi (149,5)-\varphi (\,\,\,11,5)=\,\,\,\,\,\,31\,\,\,\,\,\,\,-\,\,\,1=\,\,\,30$  $\varphi (149,5)=\varphi (149,4)-\varphi (\,\,\,13,4)=\,\,\,\,\,\,34\,\,\,\,\,\,\,-\,\,\,3=\,\,\,31$  $\varphi (149,4)=\varphi (149,3)-\varphi (\,\,\,21,3)=\,\,\,\,\,\,40\,\,\,\,\,\,\,-\,\,\,6=\,\,\,34$  $\varphi (149,3)=\varphi (149,2)-\varphi (\,\,\,29,2)=\,\,\,\,\,\,50\,\,\,\,\,\,\,-10=\,\,\,40$  $\varphi (149,2)=\varphi (149,1)-\varphi (\,\,\,49,1)=\,\,\,\,\,\,75\,\,\,\,\,\,\,-25=\,\,\,50$  $\varphi (\,\,\,29,2)=\varphi (29,1)-\varphi (\,\,\,9,1)=15-\,\,\,\,\,\,5$	$=$	$10$
$\varphi (3225,10)=\varphi (3225,9)-\varphi (\,\,\,111,9)=\,\,\,\,\,\,521\,\,\,\,\,\,\,-21=\,\,\,500$  $\varphi (3225,\,\,\,9)=\varphi (3225,8)-\varphi (\,\,\,140,8)=\,\,\,\,\,\,548\,\,\,\,\,\,\,-27=\,\,\,521$  $\varphi (3225,\,\,\,8)=\varphi (3225,7)-\varphi (\,\,\,169,7)=\,\,\,\,\,\,581\,\,\,\,\,\,\,-33=\,\,\,548$  $\varphi (3225,\,\,\,7)=\varphi (3225,6)-\varphi (\,\,\,189,6)=\,\,\,\,\,\,618\,\,\,\,\,\,\,-37=\,\,\,581$  $\varphi (3225,\,\,\,6)=\varphi (3225,5)-\varphi (\,\,\,248,5)=\,\,\,\,\,\,670\,\,\,\,\,\,\,-52=\,\,\,618$  $\varphi (3225,\,\,\,5)=\varphi (3225,4)-\varphi (\,\,\,293,4)=\,\,\,\,\,\,738\,\,\,\,\,\,\,-68=\,\,\,670$  $\varphi (3225,\,\,\,4)=\varphi (3225,3)-\varphi (\,\,\,460,3)=\,\,\,\,\,\,860\,\,\,\,-122=\,\,\,738$  $\varphi (3225,\,\,\,3)=\varphi (3225,2)-\varphi (\,\,\,645,2)=\,\,\,1075\,\,\,\,-215=\,\,\,860$  $\varphi (3225,\,\,\,2)=\varphi (3225,1)-\varphi (1075,1)=\,\,\,1613\,\,\,\,-538=1075$  $\varphi (\,\,\,645,2)=\varphi (\,\,\,645,1)-\varphi (215,1)=323-\,\,\,108=215$
$\varphi (\,\,\,460,3)=\varphi (\,\,\,460,2)-\varphi (\,\,\,92,2)=153-\,\,\,\,\,\,31=122$  $\varphi (\,\,\,460,2)=\varphi (\,\,\,460,1)-\varphi (153,1)=230-\,\,\,\,\,\,77=153$  $\varphi (\,\,\,\,\,92,2)=\varphi (\,\,\,92,1)-\varphi (30,1)=46-15$	$=$	$\,\,\,31$
$\varphi (\,\,\,293,4)=\varphi (\,\,\,293,4)-\varphi (\,\,\,41,3)=\,\,\,79-\,\,\,\,\,\,11=\,\,\,68$  $\varphi (\,\,\,293,3)=\varphi (\,\,\,293,2)-\varphi (\,\,\,58,2)=\,\,\,98-\,\,\,\,\,\,19=\,\,\,79$  $\varphi (\,\,\,293,2)=\varphi (\,\,\,293,1)-\varphi (\,\,\,97,1)=147-\,\,\,\,\,\,49=\,\,\,98$  $\varphi (\,\,\,\,\,58,2)=\varphi (\,\,\,58,1)-\varphi (19,1)=19-10$	$=$	$\,\,\,19$
$\varphi (\,\,\,41,3)\,\,\,=\,\,\,\varphi (41,2)-\varphi (\,\,\,8,2)=41-\,\,\,3$  $\varphi (\,\,\,41,2)\,\,\,=\,\,\,\varphi (41,1)-\varphi (13,1)=21-\,\,\,7$  $\varphi (\,\,\,\,\,\,8,2)\,\,\,=\,\,\,\varphi (\,\,\,8,1)-\varphi (\,\,\,2,1)=\,\,\,4-\,\,\,1$	$=$ $=$ $=$	$\,\,\,11$ $\,\,\,14$ $\,\,\,\,\,\,3$
$\varphi (\,\,\,248,5)=\varphi (\,\,\,248,4)-\varphi (\,\,\,22,4)=\,\,\,57-\,\,\,\,\,\,5=52$  $\varphi (\,\,\,248,4)=\varphi (\,\,\,248,3)-\varphi (\,\,\,35,3)=\,\,\,66-\,\,\,\,\,\,9=57$  $\varphi (\,\,\,248,3)=\varphi (\,\,\,248,2)-\varphi (\,\,\,49,2)=\,\,\,83-\,\,\,17=66$  $\varphi (\,\,\,248,2)=\varphi (\,\,\,248,1)-\varphi (\,\,\,82,1)=124-\,\,\,41=83$  $\varphi (\,\,\,49,2)\,\,\,=\,\,\,\varphi (49,1)-\varphi (16,1)=25-\,\,\,8$	$=$	$\,\,\,17$
$\varphi (\,\,\,35,3)\,\,\,=\,\,\,\varphi (35,2)-\varphi (\,\,\,7,2)=12-\,\,\,3$  $\varphi (\,\,\,35,2)\,\,\,=\,\,\,\varphi (35,1)-\varphi (11,1)=18-\,\,\,6$  $\varphi (\,\,\,\,\,\,7,2)\,\,\,=\,\,\,\varphi (\,\,\,7,1)-\varphi (\,\,\,2,1)=\,\,\,4-\,\,\,1$	$=$ $=$ $=$	$\,\,\,\,\,\,9$ $\,\,\,12$ $\,\,\,\,\,\,3$
$\varphi (\,\,\,189,6)=\varphi (\,\,\,189,5)-\varphi (\,\,\,14,5)=\,\,\,39-\,\,\,\,\,\,2=37$  $\varphi (\,\,\,189,5)=\varphi (\,\,\,189,4)-\varphi (\,\,\,17,4)=\,\,\,43-\,\,\,\,\,\,4=39$  $\varphi (\,\,\,189,4)=\varphi (\,\,\,189,3)-\varphi (\,\,\,27,3)=\,\,\,50-\,\,\,\,\,\,7=43$  $\varphi (\,\,\,189,3)=\varphi (\,\,\,189,2)-\varphi (\,\,\,37,2)=\,\,\,63-\,\,\,13=50$  $\varphi (\,\,\,189,2)=\varphi (\,\,\,189,1)-\varphi (\,\,\,63,1)=\,\,\,95-\,\,\,32=63$  $\varphi (\,\,\,37,2)\,\,\,=\,\,\,\varphi (37,1)-\varphi (12,1)=19-\,\,\,6$	$=$	$\,\,\,13$
$\varphi (\,\,\,27,3)=\varphi (\,\,\,27,2)-\varphi (\,\,\,\,\,\,5,2)=\,\,\,\,\,\,9\,\,\,\,\,\,\,-\,\,\,2=\,\,\,\,\,\,7$  $\varphi (\,\,\,27,2)=\varphi (\,\,\,27,1)-\varphi (\,\,\,\,\,\,9,1)=\,\,\,14\,\,\,\,\,\,\,-\,\,\,5=\,\,\,\,\,\,9$  $\varphi (\,\,\,5,2)=\varphi (\,\,\,5,1)-\varphi (2,1)=\,\,\,\,\,\,3-\,\,\,\,\,\,1$	$=$	$\,\,\,2$
$\varphi (\,\,\,17,4)=\varphi (\,\,\,17,3)-\varphi (\,\,\,\,\,\,1,3)=\,\,\,\,\,\,5\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,4$  $\varphi (\,\,\,17,3)=\varphi (\,\,\,17,2)-\varphi (\,\,\,\,\,\,2,2)=\,\,\,\,\,\,6\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,5$  $\varphi (\,\,\,17,2)=\varphi (\,\,\,17,1)-\varphi (\,\,\,\,\,\,5,1)=\,\,\,\,\,\,9\,\,\,\,\,\,\,-\,\,\,3=\,\,\,\,\,\,6$
$\varphi (169,7)=\varphi (169,6)-\varphi (\,\,\,\,\,\,9,6)=\,\,\,34\,\,\,\,\,\,\,-\,\,\,1=\,\,\,33$  $\varphi (169,6)=\varphi (169,5)-\varphi (\,\,\,13,5)=\,\,\,36\,\,\,\,\,\,\,-\,\,\,2=\,\,\,34$  $\varphi (169,5)=\varphi (169,4)-\varphi (\,\,\,15,4)=\,\,\,39\,\,\,\,\,\,\,-\,\,\,3=\,\,\,36$  $\varphi (169,4)=\varphi (169,3)-\varphi (\,\,\,24,3)=\,\,\,46\,\,\,\,\,\,\,-\,\,\,7=\,\,\,39$  $\varphi (169,3)=\varphi (169,2)-\varphi (\,\,\,33,2)=\,\,\,57\,\,\,\,\,\,\,-11=\,\,\,46$  $\varphi (169,2)=\varphi (169,1)-\varphi (\,\,\,56,1)=\,\,\,85\,\,\,\,\,\,\,-28=\,\,\,57$  $\varphi (33,2)=\varphi (33,1)-\varphi (11,1)=\,\,\,17-\,\,\,\,\,\,6$	$=$	$11$
$\varphi (\,\,\,24,3)=\varphi (\,\,\,24,2)-\varphi (\,\,\,\,\,\,4,2)=\,\,\,\,\,\,8\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,7$  $\varphi (\,\,\,24,2)=\varphi (\,\,\,24,1)-\varphi (\,\,\,\,\,\,8,1)=\,\,\,12\,\,\,\,\,\,\,-\,\,\,4=\,\,\,\,\,\,8$
$\varphi (140,8)=\varphi (140,7)-\varphi (\,\,\,\,\,\,7,7)=\,\,\,28\,\,\,\,\,\,\,-\,\,\,1=\,\,\,27$  $\varphi (140,7)=\varphi (140,6)-\varphi (\,\,\,\,\,\,8,6)=\,\,\,29\,\,\,\,\,\,\,-\,\,\,1=\,\,\,28$  $\varphi (140,6)=\varphi (140,5)-\varphi (\,\,\,10,5)=\,\,\,30\,\,\,\,\,\,\,-\,\,\,1=\,\,\,29$  $\varphi (140,5)=\varphi (140,4)-\varphi (\,\,\,12,4)=\,\,\,32\,\,\,\,\,\,\,-\,\,\,2=\,\,\,30$  $\varphi (140,4)=\varphi (140,3)-\varphi (\,\,\,20,3)=\,\,\,38\,\,\,\,\,\,\,-\,\,\,6=\,\,\,32$  $\varphi (140,3)=\varphi (140,2)-\varphi (\,\,\,28,2)=\,\,\,47\,\,\,\,\,\,\,-\,\,\,9=\,\,\,38$  $\varphi (140,2)=\varphi (140,1)-\varphi (\,\,\,46,1)=\,\,\,70\,\,\,\,\,\,\,-23=\,\,\,47$  $\varphi (\,\,\,28,2)=\varphi (\,\,\,28,1)-\varphi (\,\,\,\,\,\,9,1)=\,\,\,14\,\,\,\,\,\,\,-\,\,\,5=\,\,\,\,\,\,9$
$\varphi (\,\,\,20,3)=\varphi (\,\,\,20,2)-\varphi (\,\,\,\,\,\,4,2)=\,\,\,\,\,\,7\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,6$  $\varphi (\,\,\,20,2)=\varphi (\,\,\,20,1)-\varphi (\,\,\,\,\,\,6,1)=\,\,\,10\,\,\,\,\,\,\,-\,\,\,3=\,\,\,\,\,\,7$  $\varphi (\,\,\,4,2)=\varphi (\,\,\,4,1)-\varphi (\,\,\,1,1)=\,\,\,\,\,\,2-\,\,\,\,\,\,1$	$=$	$\,\,\,1$
$\varphi (\,\,\,12,4)=\varphi (\,\,\,12,3)-\varphi (\,\,\,\,\,\,1,3)=\,\,\,\,\,\,3\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,2$  $\varphi (\,\,\,12,3)=\varphi (\,\,\,12,2)-\varphi (\,\,\,\,\,\,2,2)=\,\,\,\,\,\,4\,\,\,\,\,\,\,-\,\,\,1=\,\,\,\,\,\,3$  $\varphi (\,\,\,12,2)=\varphi (\,\,\,12,1)-\varphi (\,\,\,\,\,\,4,1)=\,\,\,\,\,\,6\,\,\,\,\,\,\,-\,\,\,2=\,\,\,\,\,\,4$
$\varphi (111,9)=\varphi (111,8)-\varphi (\,\,\,\,\,\,4,8)=\,\,\,22\,\,\,\,\,\,\,-\,\,\,1=\,\,\,21$  $\varphi (111,8)=\varphi (111,7)-\varphi (\,\,\,\,\,\,5,7)=\,\,\,23\,\,\,\,\,\,\,-\,\,\,1=\,\,\,22$  $\varphi (111,7)=\varphi (111,6)-\varphi (\,\,\,\,\,\,6,6)=\,\,\,24\,\,\,\,\,\,\,-\,\,\,1=\,\,\,23$  $\varphi (111,6)=\varphi (111,5)-\varphi (\,\,\,\,\,\,8,5)=\,\,\,25\,\,\,\,\,\,\,-\,\,\,1=\,\,\,24$  $\varphi (111,5)=\varphi (111,4)-\varphi (\,\,\,10,4)=\,\,\,26\,\,\,\,\,\,\,-\,\,\,1=\,\,\,25$  $\varphi (111,4)=\varphi (111,3)-\varphi (\,\,\,15,3)=\,\,\,30\,\,\,\,\,\,\,-\,\,\,4=\,\,\,26$  $\varphi (111,3)=\varphi (111,2)-\varphi (\,\,\,22,2)=\,\,\,37\,\,\,\,\,\,\,-\,\,\,7=\,\,\,30$  $\varphi (111,2)=\varphi (111,1)-\varphi (\,\,\,37,1)=\,\,\,56\,\,\,\,\,\,\,-19=\,\,\,37$  $\varphi (22,2)=\varphi (22,1)-\varphi (\,\,\,7,1)=\,\,\,11-\,\,\,\,\,\,4$	$=$	$\,\,\,7$
$\varphi (2702,11)=\varphi (2702,10)-\varphi (\,\,\,87,10)=\,\,\,\,\,\,420\,\,\,\,\,\,\,-\,\,\,14=\,\,\,406$  $\varphi (2702,10)=\varphi (2702,\,\,\,9)-\varphi (\,\,\,93,\,\,\,9)=\,\,\,\,\,\,436\,\,\,\,\,\,\,-\,\,\,16=\,\,\,420$  $\varphi (2702,\,\,\,9)=\varphi (2702,\,\,\,8)-\varphi (117,\,\,\,8)=\,\,\,\,\,\,459\,\,\,\,\,\,\,-\,\,\,23=\,\,\,436$  $\varphi (2702,\,\,\,8)=\varphi (2702,\,\,\,7)-\varphi (142,\,\,\,7)=\,\,\,\,\,\,487\,\,\,\,\,\,\,-\,\,\,28=\,\,\,459$  $\varphi (2702,\,\,\,7)=\varphi (2702,\,\,\,6)-\varphi (158,\,\,\,6)=\,\,\,\,\,\,519\,\,\,\,\,\,\,-\,\,\,32=\,\,\,487$  $\varphi (2702,\,\,\,6)=\varphi (2702,\,\,\,5)-\varphi (207,\,\,\,5)=\,\,\,\,\,\,562\,\,\,\,\,\,\,-\,\,\,43=\,\,\,519$  $\varphi (2702,\,\,\,5)=\varphi (2702,\,\,\,4)-\varphi (245,\,\,\,4)=\,\,\,\,\,\,618\,\,\,\,\,\,\,-\,\,\,56=\,\,\,562$  $\varphi (2702,\,\,\,4)=\varphi (2702,\,\,\,3)-\varphi (386,\,\,\,3)=\,\,\,\,\,\,721\,\,\,\,\,\,\,-103=\,\,\,618$  $\varphi (2702,\,\,\,3)=\varphi (2702,\,\,\,2)-\varphi (540,\,\,\,2)=\,\,\,\,\,\,901\,\,\,\,\,\,\,-180=\,\,\,721$  $\varphi (2702,\,\,\,2)=\varphi (2702,\,\,\,1)-\varphi (900,\,\,\,1)=\,\,\,1351\,\,\,\,\,\,\,-450=\,\,\,901$  $\varphi (\,\,\,540,\,\,\,2)=\varphi (540,\,\,\,1)-\varphi (180,1)=270-\,\,\,\,\,\,90$	$=$	$180$
$\varphi (386,\,\,\,3)=\varphi (386,\,\,\,2)-\varphi (\,\,\,77,2)=129-\,\,\,\,\,\,26$  $\varphi (386,\,\,\,2)=\varphi (386,\,\,\,1)-\varphi (128,1)=193-\,\,\,\,\,\,64$  $\varphi (\,\,\,77,\,\,\,2)=\varphi (\,\,\,77,\,\,\,1)-\varphi (\,\,\,25,1)=\,\,\,39-\,\,\,\,\,\,13$	$=$ $=$ $=$	$103$ $129$ $\,\,\,26$
$\varphi (245,\,\,\,4)=\varphi (245,\,\,\,3)-\varphi (\,\,\,35,3)=\,\,\,65-\,\,\,\,\,\,9$  $\varphi (245,\,\,\,3)=\varphi (245,\,\,\,2)-\varphi (\,\,\,49,2)=\,\,\,82-\,\,\,17$  $\varphi (245,\,\,\,2)=\varphi (245,\,\,\,1)-\varphi (\,\,\,81,1)=123-\,\,\,41$  $\varphi (49,\,\,\,2)=\varphi (49,\,\,\,1)-\varphi (\,\,\,16,1)=\,\,\,25-\,\,\,\,\,\,8$	$=$ $=$ $=$ $=$	$\,\,\,56$ $\,\,\,65$ $\,\,\,82$ $\,\,\,17$
$\varphi (\,\,\,35,\,\,\,3)=\varphi (\,\,\,35,\,\,\,2)-\varphi (\,\,\,\,\,\,7,2)=\,\,\,12-\,\,\,\,\,\,3$  $\varphi (\,\,\,35,\,\,\,2)=\varphi (\,\,\,35,\,\,\,1)-\varphi (\,\,\,11,1)=\,\,\,18-\,\,\,\,\,\,6$  $\varphi (7,\,\,\,2)=\varphi (\,\,\,7,\,\,\,1)-\varphi (\,\,\,2,1)=\,\,\,\,\,\,4-\,\,\,\,\,\,1$	$=$ $=$ $=$	$\,\,\,\,\,\,9$ $\,\,\,12$ $\,\,\,\,\,\,3$
$\varphi (207,\,\,\,5)=\varphi (207,\,\,\,4)-\varphi (\,\,\,18,4)=\,\,\,47-\,\,\,\,\,\,4$  $\varphi (207,\,\,\,4)=\varphi (207,\,\,\,3)-\varphi (\,\,\,29,3)=\,\,\,55-\,\,\,\,\,\,8$  $\varphi (207,\,\,\,3)=\varphi (207,\,\,\,2)-\varphi (\,\,\,41,2)=\,\,\,69-\,\,\,14$  $\varphi (207,\,\,\,2)=\varphi (207,\,\,\,1)-\varphi (\,\,\,69,1)=104-\,\,\,35$  $\varphi (41,\,\,\,2)=\varphi (\,\,\,41,\,\,\,1)-\varphi (\,\,\,13,1)=\,\,\,21-\,\,\,\,\,\,7$	$=$ $=$ $=$ $=$ $=$	$\,\,\,43$ $\,\,\,47$ $\,\,\,55$ $\,\,\,69$ $\,\,\,14$
$\varphi (29,\,\,\,3)=\varphi (\,\,\,29,\,\,\,2)-\varphi (\,\,\,\,\,\,5,2)=\,\,\,10-\,\,\,\,\,\,2$  $\varphi (29,\,\,\,2)=\varphi (\,\,\,29,\,\,\,1)-\varphi (\,\,\,\,\,\,9,2)=\,\,\,15-\,\,\,\,\,\,5$  $\varphi (\,\,\,5,\,\,\,2)=\varphi (\,\,\,\,\,\,5,\,\,\,1)-\varphi (1,1)=\,\,\,\,\,\,3-\,\,\,\,\,\,1$	$=$ $=$ $=$	$\,\,\,\,\,\,8$ $\,\,\,10$ $\,\,\,\,\,\,2$
$\varphi (18,\,\,\,4)=\varphi (\,\,\,18,\,\,\,3)-\varphi (\,\,\,2,3)=\,\,\,\,\,\,5-\,\,\,\,\,\,1$  $\varphi (18,\,\,\,3)=\varphi (\,\,\,18,\,\,\,2)-\varphi (\,\,\,3,2)=\,\,\,\,\,\,6-\,\,\,\,\,\,1$  $\varphi (18,\,\,\,2)=\varphi (\,\,\,18,\,\,\,1)-\varphi (\,\,\,6,1)=\,\,\,\,\,\,9-\,\,\,\,\,\,3$	$=$ $=$ $=$	$\,\,\,\,\,\,4$ $\,\,\,\,\,\,5$ $\,\,\,\,\,\,6$
$\varphi (158,\,\,\,6)=\varphi (158,\,\,\,5)-\varphi (\,\,\,12,5)=\,\,\,33-\,\,\,\,\,\,1=\,\,\,32$  $\varphi (158,\,\,\,5)=\varphi (158,\,\,\,4)-\varphi (\,\,\,14,4)=\,\,\,36-\,\,\,\,\,\,3=\,\,\,33$  $\varphi (158,\,\,\,4)=\varphi (158,\,\,\,3)-\varphi (\,\,\,22,3)=\,\,\,42-\,\,\,\,\,\,6=\,\,\,36$  [18]$\varphi (158,\,\,\,3)=\varphi (158,\,\,\,2)-\varphi (\,\,\,31,2)=\,\,\,53-\,\,\,11=\,\,\,42$  $\varphi (158,\,\,\,2)=\varphi (158,\,\,\,1)-\varphi (\,\,\,52,1)=\,\,\,79-\,\,\,26=\,\,\,53$  $\varphi (\,\,\,31,\,\,\,2)=\varphi (31,\,\,\,1)-\varphi (10,1)=\,\,\,16-\,\,\,\,\,\,5$	$=$	$\,\,\,11$
$\varphi (\,\,\,22,\,\,\,3)=\varphi (\,\,\,22,\,\,\,2)-\varphi (\,\,\,\,\,\,4,2)=\,\,\,\,\,\,7-\,\,\,\,\,\,1=\,\,\,\,\,\,6$  $\varphi (\,\,\,22,\,\,\,2)=\varphi (\,\,\,22,\,\,\,1)-\varphi (\,\,\,\,\,\,7,1)=\,\,\,11-\,\,\,\,\,\,4=\,\,\,\,\,\,7$  $\varphi (4,\,\,\,2)=\varphi (\,\,\,4,\,\,\,1)-\varphi (1,1)=\,\,\,\,\,2-\,\,\,\,\,\,1$	$=$	$\,\,\,\,\,\,1$
$\varphi (\,\,\,14,\,\,\,4)=\varphi (\,\,\,14,\,\,\,3)-\varphi (\,\,\,\,\,\,2,3)=\,\,\,\,\,\,4-\,\,\,\,\,\,1=\,\,\,\,\,\,3$  $\varphi (\,\,\,14,\,\,\,3)=\varphi (\,\,\,14,\,\,\,2)-\varphi (\,\,\,\,\,\,2,2)=\,\,\,\,\,\,5-\,\,\,\,\,\,1=\,\,\,\,\,\,4$  $\mathrm{\phi <!--\; \phi \; -->}(14,2)=\mathrm{\phi <!--\; \phi \; -->}(14,1)-<!--\; -\; -->\mathrm{\phi <!--\; \phi \; -->}(4,1)=7-<!--\; -\; -->2=$