The Theory of Numbers
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, m r by the relations m = dimi=d2m2= . . . =d T m r . Now consider the set of m positive integers not greater than m, and classify them as follows into r classes. Place in the first class those integers of the set which have with m the great- est common divisor mi; -their number is(di), as may be seen from the lemma. Place in the second class those integers of the set which have with m the greatest common divisor m 2 \ their number is 4>(d 2 )- Proceeding in this way throughout, we place fina ...lly in the last class those integers of the set which have with m the greatest common divisor m r ', their number is 4>{d r ). It is evident that every integer in the set falls into one and into just one of these r classes. Hence the total num- ber m of integers in the set is 4>{d\) -\- 4>{d 2 ) -{- . . . -\-(d r ). From this the theorem follows immediately. 36 THEORY OF NUMBERS EXERCISES 1. Show that the indicator of any integer greater than 2 is even. 2. Prove that the number of irreducible fractions not greater than 1 and with denominator equal to n is (n).
What to read after The Theory of Numbers?
You can find similar books in the "Read Also" column, or choose other free books by R D Robert Daniel Carmichael to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
, m r by the relations m = dimi=d2m2= . . . =d T m r . Now consider the set of m positive integers not greater than m, and classify them as follows into r classes. Place in the first class those integers of the set which have with m the great- est common divisor mi; -their number is
What to read after The Theory of Numbers?
You can find similar books in the "Read Also" column, or choose other free books by R D Robert Daniel Carmichael to read onlineMoreLess
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