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EXAMPLES FOR PRACTICE. 2. Find all the primes from 1 up to 127. 3. Find all the primes from 1 up to 181. 4. Find all the primes below 300. 5. Find all the primes between 300 and 400.

CASE II.

962. To ascertain if a given number is prime.

Rule.-I. Search for the number in the table, if contained within its limits; if it is found there it is prime, if not, it is composite.

II. Divide the number by the successive primes; if an exact divisor is found, the number is composite; if we continue the division until the quotient is less than the divisor without finding an exact divisor, the number is prime.

NOTE.—The Table of Prime Numbers will be found on the following page.

EXAMPLES FOR PRACTICE. Determine which of the following numbers are prime : 1. 273.

5. 3413. 2. 649.

6. 3853. 3. 2671.

7. 4001. 4. 3063.

8. 4049.

NOTES.—1. Several remarkable formulas have been discovered, which contain many prime numbers. Thus, the formula x2+x+41, by making successively x=0, 1, 2, 3, 4, etc., will give the series 41, 43, 47, 53, 61, 71, etc., the first forty terms of which are prime numbers. This formula is mentioned by Euler.

2. The formula x2+x+17 gives seventeen of its first terms prime; and the formula 2x2+29 gives twenty-nine of its first terms prime. Fermat asserted that the formula 2m +1 is always a prime when m is taken any term in the series 1, 2, 4, 8, 16, etc.; but Euler found that 232+1, which equals 641X 6700417, is not a prime.

3. One of the most celebrated theorems for investigating primes is that discovered by Fermat, known as Fermat's Theorem. This formula may be stated thus: If p be a prime number, the (p-1)th power of every number prime to p will, when diminished by unity, be exactly divisible by p. Thus 256 — 1 is exactly divisible by 7. For a fuller discussion of the subject, see the author's Philosophy of Arithmelic.

TABLE OF PRIME NUMBERS. 963. A Table of Prime Numbers is a list of the prime numbers from 1 up to any given limit.

961. The following table contains the prime numbers from 1 up to 3407.

TABLE OF PRIMES.

1/173 403 659 941 1223 1511 1811 2129 2423 2741 3079 2 179 419.661 917 1229 1523 1823 2131 2437 2749 3083 3 181421 1 673 9:53 1231 1531 / 1831 2137 2441 2753 3089 5.191 431 677 957 1237 | 1543 1847 2141 2447 2767 3109 7 193 133 633 971 1249 1519 1861 2143 2459 2777 3119 11197 439 691 977 1259 1553 1867 2153 2467 2789 3121 13 199 443 701 983 1277 1559 1871 2161 2473 2791 3137 17211 449 709 991 1279 | 1567 1873 2179 2477 2797 3163 19 223 457 1 719997 1233 1571 1877 2203 2503 2801 3167 23 227 461 727 1009 1289 1579 1879 22072521 2803 3169 29 229 463 733 1013 1291 1583 1889 2213 2531 2819 3181 31 233 467 739 1019 1297 1597 1901 2221 2539 2833 3187 37 239 479 743 1021 1301 1601 1907 2237 2543 | 2837 3191 41 241 487 751 1031 1303 1607 1913 2239 2549 2843 3203 43 251 491 757 1033 1307 1609 1931 2243 2551 2851 | 3209 47 257 499 761 1039 1319 1613 1933 2251 2557 2857 3217 53 263 503 769 1019 1321 1619 1949 2267 2579 2861 3221 59 269 503 773 1051 1327 1621 1951 2269 2591 2879 3229 61 271 521 787 1061 1361 1627 1973 2271 2593 2887 3251 67 277 523 797 1063 1367 1637 | 1979 2283 2609 2897 3253 71 281 541 809 1069 1373 1657 1937 2287 2617 2903 3257 73 283 547 811 1087 1331 1663 1993 2293 2621 2909 3259 79293 557 821 1091 1399 1667 1997 2297 2633 2917 3271 83 307 563 823 1093 1409 1669 1999 2309 26 17 2927 3299 89 311 569 827 1097 1423 1693 2003 2311 2657 2939 3301 97 313 571 829 1103 1427 1697 2011 2333 2659 2953 3307 101 317 577 839 1109 1429 1699 2017 2339 2663 2957 3313 103331 587 | 853 1117 1433 1709 | 2027 2341 2671 2963 3319 107 337 593 857 1123 1439 1721 2029 2317 2677 2969 3323 109 317 599 859 1129 1447 1723 2039 2351 2683 2971 | 3329 113 319 601 863 1151 1451 1733 2053 2357 2687 2999 3331 127 353 607 877 1153 1453 1741 2063 2371 2689 3001 3343 131 359 613 881 11631459 1747 2069 2377 2693 3011 | 3347 137 367 617 883 11711471 1753 2081 2381 2699 3019 3359 139 373 619 837 1181 1481 1759 2033 2383 2707 3023 3361 149 379 631 / 907 1187 | 1483 1777 2087 2339 2711 3037 3371 151 383 641 911 1193 1487 1783/2039 2393 2713 3041 | 3373 157 | 389 643 919 1201 1439 1787 2099 2399 2719 3049 3389 163397 647 929 1213 1493 1789 2111 2411 2729 | 3061 3391 167 401|653 937|12:7 1499|1801 / 2113 2417 2731 3067 3407

EVEN AND ODD NUMBERS. 965. An Even Number is one that is exactly divisible by 2; as, 2, 4, 6, etc.

966. An Odd Number is one that is not exactly divisible by 2; as, 1, 3, 5, 7, etc.

967. The Even Numbers are divided into the oddly even numbers, as 2, 6, 10, 14, etc., and the evenly even numbers, as 4, 8, 12, 16, etc.

968. The Odd Numbers are divided into the evenly odd numbers, as 1, 5, 9, 13, etc., and the oddly odd numbers, as 3, 7, 11, 15, etc.

NOTE.— The form of an even number is 2n; the form of an odd number is 2n+1, in which n represents any integer. In the evenly even numbers, n (in 2n) is even ; in the oddly even numbers, n is odd. In the evenly odd, 10 (in 2n+1) is even ; in the oddly odd, n is odd.

PRINCIPLES. 1. Every even number equals a NUMBER OF 2's, and every odd number equals & NUMBER OF 2’s, plus 1.

For, since an even number is divisible by 2, it is evidently equal to a number of 2's; and since an odd number is not exactly divisible by 2, there will be a remainder of 1; hence an odd number equals a number of 2's, plus 1.

2. The sum or difference of two even numbers is even. For, since both numbers equal a number of 2's, their sum is a number of 2's plus another number of 2's, which equals a number of 2's; hence the sum is an even number. Their difference equals a number of 2's minus another number of 2's, which equals a number of 2's; hence their difference is an even number.

3. The sum or difference of two odd numbers is even.

For, each number equals a number of 2's,+1, hence their sum equals a number of 2's,+2, or an exact number of 2's; hence their sum is even. Their difference equals an exact number of 2's; hence their difference is

even,

4. The sum or difference of an even number and an odd number is odd.

For, the even number equals a number of 2's, and the odd number equals a number of 2's,+1; hence their sum and difference will equal a number of 2's, +1, and be an odd number.

5. The product of two even numbers is an even number.

For, since both of them contain the factor 2, their product will contain the factor 2, and therefore be even.

6. The product of two odd numbers is an odd number.

For, since neither of them contains the factor 2, their product will not contain the factor 2, and will therefore be odd.

7. The product of an even and an odd number is an even number.

For, since one of the numbers contains the factor 2, the product of the two numbers will contain the factor 2, and will therefore be even.

8. If an even number is exactly divisible by an odd number, the quotient will be even.

For, the divisor multiplied by the quotient equals the dividend, hence when the dividend is even and the divisor odd, the quotient must be even, since an odd number multiplied by an even number will give an even number.

9. If an odd number is exactly divisible by an odd number, the quotient is odd.

For, since an odd number must be multiplied by an odd number to produce an odd number, the quotient must be odd that the product of it and the divisor may equal the odd dividend.

10. If an even number is exactly divisible by an even number, the quotient may be even or odd.

For, an even number multiplied by either an even or an odd number will produce an even number, hence the quotient may be even or odd.

11. An odd number is not exactly divisible by an even number, and the remainder is odd.

Since an even number multiplied by no integral number will produce an odd number, an odd number is not exactly divisible by an even number. The remainder is odd, since it is the difference between an odd number and an even number.

12. If an even number is not exactly divisible by another even number, the remainder is even.

For, the remainder will be the difference between the dividend and a number of times the divisor, that is, the difference between two even numbers, which is even.

13. If an even number is not exactly divisible by an odd number, then when the quotient is even the remainder is even, and when the quotient is odd the remainder is odd.

14. If an odd number is not exactly divisible by an odd number, then when the quotient is odd the remainder is even, and when the quotient is even the remainder is odd.

NOTE.—Let the pupil be required to demonstrate the last two principles.

PERFECT AND IMPERFECT NUMBERS. 969. A Perfect Number is one which is equal to the sum of all its divisors except itself; thus, 6=1+2+3; 28= 1+2+4+7+14.

970. An Imperfect Number is one which is not equal to the sum of all its divisors; Imperfect Numbers are Abundant or Defective.

971. An Abundant Number is one the sum of whose divisors exceeds the number itself; as, 18<1+2+3+6+9.

972. A Defective Number is one the sum of whose divisors is less than the number itself; as, 16>1+2+4+8.

973. Two numbers are called Amicable Numbers, when each is equal to the sum of the divisors of the other; thus, 284 and 220.

Notes.—1. Every number of the form (2n-1)(2n--1), the latter factor being a prime number, is a perfect number. The difficulty in finding perfect numbers consists in finding primes of the form of 2n--1, which is very laborious. Substituting 2 for n in the formula just given, we have 2X. (22-1)=6, the first perfect number; the second is 22 X (23--1)=28.

2. The following are the first eight perfect numbers : 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128. It will be noticed that each number ends in 6 or 28.

3. The formulas for finding amicable numbers are A=2n+1 dand B=2n+1bc, in which n is an integer and b, c, and d are prime numbers satisfying the following conditions:

1st, b=3X20--1; 2d, c=6X2n—-1; 3d, d=18 X 221_-1. If we maken=1 we find b = 5, c=11, d=71'; substituting these in the above formulas, we have A=4X71=284, and B=4X5X11=220, the first pair of amicable numbers. The next two pairs are 17296, 18416, and 936358, 9437056.

4. Figurate Numbers are formed from an arithmetical progression whose first term is unity, and common difference integral, by taking successively the sum of the first two, the first three, the first four, etc., terms; and then forming in the same manner another series from the one just obtained and so on. For a discussion of Figurate Numbers, see Philosophy of Arithmetic.

EXAMPLES FOR PRACTICE. 1. Find the third perfect number by the formula (n=5). 2. Find the fourth perfect number by the formula (n=7). 3. Show that 496 and 8128 are perfect numbers. 4. Find the second pair of amicable numbers (n=3). 5. Show that 220 and 284 are amicable numbers. 6. Sbow that 17296 and 18416 are amicable numbers.

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