Math 3221 Number Theory - Homework Until Test 2 Worksheet With Answers - Philipp Braun Page 6
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1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16other has to be odd. Since the only even prime is 2, the only of these pairs is p = 5 and
q = 2.
5. In 1752, Goldbach submitted the following conjecture to Euler: Every odd integer can be
2
written in the form p + 2a
, were p is either a prime or 1 and a
0. Show that the integer 5777
refutes this conjecture.
2
Solution. a has to be less than or equal to 53, since 2 54
= 5832 > 5777. The
2
set 5777
2a
0
a
53 = 5777 = 53 109, 5775 = 5 1155, 5769 = 3 1923, 5759 =
13 443, 5745 = 5 1149, 5727 = 3 1909, 5705 = 5 1141, 5679 = 3 1893, 5649 = 3 1883, 5615 =
5 1123, 5577 = 3 1859, 5535 = 5 1107, 5489 = 11 499, 5439 = 3 1813, 5385 = 5 1077, 5327 =
7 761, 5265 = 5 1053, 5199 = 3 1733, 5129 = 23 223, 5055 = 5 1011, 4977 = 3 1659, 4895 =
5 979, 4809 = 3 1603, 4719 = 3 1573, 4625 = 5 925, 4527 = 3 1509, 4425 = 5 885, 4319 =
7 617, 4209 = 3 1403, 4095 = 5 819, 3977 = 41 97, 3855 = 5 771, 3729 = 3 1243, 3599 =
59 61, 3465 = 5 693, 3327 = 3 1109, 3185 = 5 637, 3039 = 3 1013, 2889 = 3 963, 2735 =
5 547, 2577 = 3 859, 2415 = 5 483, 2249 = 13 173, 2079 = 3 693, 1905 = 5 381, 1727 =
11 157, 1545 = 5 309, 1359 = 3 453, 1169 = 7 167, 975 = 5 195, 777 = 7 111, 575 =
5 115, 369 = 3 123, 159 = 3 53 contains neither a prime nor 1, so 5777 cannot be written
in the wanted form.
7. A conjecture of Lagrange (1775) asserts that every odd integer greater than 5 can be written
as a sum p
+ 2p
, where p
, p
are both primes. Confirm this for all odd integers through 75.
1
2
1
2
Solution. 7 = 3 + 2 2, 9 = 5 + 2 2, 11 = 7 + 2 2, 13 = 7 + 2 3, 15 = 5 + 2 5, 17 =
7 + 2 5, 19 = 5 + 2 7, 21 = 7 + 2 7, 23 = 13 + 2 5, 25 = 3 + 2 11, 27 = 5 + 2 11, 29 =
7 + 2 11, 31 = 5 + 2 13, 33 = 7 + 2 13, 35 = 13 + 2 11, 37 = 11 + 2 13, 39 = 13 + 2 13, 41 =
7 + 2 17, 43 = 5 + 2 19, 45 = 7 + 2 19, 47 = 13 + 2 17, 49 = 3 + 2 23, 51 = 5 + 2 23, 53 =
7 + 2 23, 55 = 17 + 2 19, 57 = 19 + 2 19, 59 = 37 + 2 11, 61 = 3 + 2 29, 63 = 5 + 2 29, 65 =
7 + 2 29, 67 = 5 + 2 31, 69 = 7 + 2 31, 71 = 13 + 2 29, 73 = 47 + 2 13, 75 = 13 + 2 31.
6
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