a, b, c ∈ Z v c 6= 0, J|ìbnyJtÝÃÍP²

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Exercise

Chapter 1. JóÝÃÍP²

(1) ' a, b, c ∈ Z v c 6= 0, J|ìbnyJtÝÃÍP².

(a) u c | a + b v c | a, J c | b.

(b) a | buv°u ac | bc.

(c) u a | b, JE n ∈ N /b aⁿ| bⁿ.

(d) u a > 1 v a − 1 | b − 1 |C a − 1 | bc − 1, J a − 1 | c − 1.

(2) ' a, m, n ∈ N v a > 1, |ì&ÆJ a^m− 1 | aⁿ− 1 uv°u m | n.

(a) ¿à.P5 x^h− 1 = (x − 1)(x^h−1+ x^h−2+ · · · + x + 1) (Í h ∈ N), Ju m | n, J a^m− 1 | aⁿ− 1.

(3) |ì&Æ+ÛËË!PÝ division algorithm (t°æ§). 9&ÆG

' a, b ∈ Z v b 6= 0 (Ä' b ∈ N).

(a) JD3°×Ý h, r ∈ Z 

a = bh + r v 0 ≤ r < |b| . (b) JD3°×Ý h, r ∈ Z 

a = bh + r v − |b|

2 < r ≤ |b|

2 .

(4) ' a, b ∈ Z v d Î/) {ma + nb | m, n ∈ Z} tÝÑJó. J

u r ∈ N, J rd Î/) {mra + nrb | m, n ∈ Z} tÝÑJó. µhJ

 gcd(ra, rb) = r gcd(a, b).

———————————– 06 March, 2008

1

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(5) ' a, b, c ∈ Z, ¿à§

5gcd(a, b) = 1 uv°uD3 r, s ∈ Z ¸ÿ ra + sb = 16 J|ìbny!²ÝP².

(a) ' gcd(a, b) = 1 v c | a + b. J gcd(a, c) = gcd(b, c) = 1.

(b) ' m, n ∈ N. J gcd(a, b) = 1 uv°u gcd(a^m, bⁿ) = 1.

(6) ' a, b ∈ Z v d = gcd(a + b, a − b).

(a) J d | 2a v d | 2b.

(b) uá gcd(a, b) = 1 J a, b ! óT! ó` d = 2;  a, b ××` d = 1.

(7) ' n ∈ N, n > 2 v a1, a₂, . . . , a_n∈ Z.

(a) E t ∈ N v 1 ≤ t ≤ n − 1, J

gcd(a₁, . . . , a_n) = gcd(gcd(a₁, . . . , a_t), gcd(a_t+1, . . . , a_n)).

(b) E t1, . . . , t_r ∈ NÍ t0 = 0 < t₁ < t₂ < · · · < t_r < n = t_r+1,u E 0 ≤ i ≤ r, di = gcd(a_t_i₊₁, . . . , a_t_i+1). J

gcd(a₁, . . . , a_n) = gcd(d₀, d₁, . . . , d_r).

(8) ' n ∈ N, n > 2 v a1, a₂, . . . , a_n∈ Z.

(a) E t ∈ N v 1 ≤ t ≤ n − 1, J

lcm(a₁, . . . , a_n) = lcm(lcm(a₁, . . . , a_t), lcm(a_t+1, . . . , a_n)).

(b) E t1, . . . , t_r ∈ NÍ t0 = 0 < t₁ < t₂ < · · · < t_r < n = t_r+1,u E 0 ≤ i ≤ r, li = lcm(a_t_i₊₁, . . . , a_t_i+1). J

lcm(a₁, . . . , a_n) = lcm(l₀, l₁, . . . , l_r).

(9) ' a1, a₂, . . . , a_n∈ N. J lcm(a1, a₂, . . . , a_n) = a₁a₂· · · a_n uv°u a₁, a₂, . . . , a_n ËË!² (pairwise relatively prime).

———————————– 13 March, 2008

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Chapter 1. JóÝÃÍP² 3

(10) ' a, b, c ∈ Z v d = gcd(a, b).

(a) J]P ax + by = c bJóuv°u d | c.

(b) ' d | c v x = m0, y = n₀ Î ax + by = d Ý×àJó, ¶ì ax + by = cÝXbJó (à a, b, c, d, m0, n₀ î).

(11) ¶|ì diophantine equations ÝXbJó.

(a) 18x + 27y = 15.

(b) 17x + 29y = 10.

(12) ' a1, a₂, . . . , a_n∈ Z v d = gcd(a1, a₂, . . . , a_n).

(a) u c ∈ Z v d - c, J]P

a₁x₁+ a₂x₂+ · · · + a_nx_n = c PJó.

(b) u c ∈ Z v d | c, J]P

a₁x₁+ a₂x₂+ · · · + a_nx_n = c bP§9àJó.

(13) ¶|ì diophantine equations ÝXbJó.

(a) 9x₁+ 12x₂+ 16x₃= 13.

(b) 8x₁− 4x₂+ 6x₃ = 6.

(14) ' a, b ∈ Z, J gcd(a, b) = 1 uv°u gcd(a + b, ab) = 1.

(Hint: ¿à Euclid Ý Lemma |Cy 1 ÝJó/b².ó§)

———————————– 20 March, 2008

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(15) 0Ý×à a, b ∈ N gcd(a, b) = 12 v lcm(a, b) = 360.

(16) ' a, b, n ∈ N uá ab = n² v gcd(a, b) = 1, JD3 c, d ∈ N 

 a = c² v b = d².

(17) ' m ∈ N v p Î²ó, A pâ | mv pâ+1- m,J&ÆÌ pâ ªJt m và pâ||m î. ¨' pâ||m v p^b||n.

(a) uá a < b, O r p^r||m + n.

(b) Ü×Í a = b Ý»¸ÿ p^r||m + nv r > a.

(c) O s p^s||mn.

Chapter 2. Arithmetic Function

(1) &ÆL×Í arithmetic function ρ ρ(1) = 1 vE n > 1 L ρ(n) = 2^m Í m n Ý8²².óÍó.

(a) J ρ Î multiplicative v1 ρ Î completely multiplicative.

(b)

f (n) = X

d|n,d∈N

ρ(d).

u n = pⁿ₁¹· · · pⁿ_r^r n Ý².ó5, O f(n).

(2) XÛÝ Liouville λ-function Î×Í arithmetic function λ ÍLAì:

λ(1) = 1 vE n > 1 u n Ý².ó5 n = pⁿ₁¹· · · pⁿ_r^r,J λ(n) = (−1)ⁿ¹^+···+n^r.

(a) J λ Î completely multiplicative.

(b)

F (n) = X

d|n,d∈N

λ(d),

JAD3 a ∈ N ¸ÿ n = a²,J F (n) = 1; ÍJ F (n) = 0.

———————————– 27 March, 2008

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Chapter 3. Congruences 5

(3) |ìÎ¿Íny Euler φ-function ÝP². h m, n ∈ N.

(a) ' n = pⁿ₁¹· · · pⁿ_r^r Î n Ý².ó5. J

φ(n) =¡

pⁿ₁¹⁻¹· · · pⁿ_r^r⁻¹¢¡

(p₁− 1) · · · (p_r− 1)¢ .

¬µhJ √

n/2 ≤ φ(n) ≤ n.

(b) Ju n óJ φ(2n) = φ(n), u n óJ φ(2n) = 2φ(n).

(c) ' n b m Í8²².ó, J 2^m | φ(n).

(d) J φ(n^m) = n^m−1φ(n).

(e) ' m | n, J φ(m) | φ(n) v φ(mn) = mφ(n).

(4) á f, g / multiplicative arithmetic function. váE²ó p |C m ∈ N/b f(p^m) = p + 1v g(p^m) = p^m−1.

(a) u f ∗ g î f õ g Ý convolution, O f ∗ g(100) Â.

(b) uE n ∈ N /b g(n) =P

d|n,d∈Nh(d). O h(100) Â.

Chapter 3. Congruences

(1) ' a, b, c, d ∈ Z v m ∈ N Í c ≡ d (mod m).

(a) á gcd(c, m) = 1, u ac ≡ bd (mod m), J a ≡ b (mod m).

(b) u c õ m !², 0×D» ac ≡ bd (mod m) ¬ a 6≡ b (mod m).

(2) ' n ∈ N v n Ý 10 î abcabc, Í a, b, c ∈ N v 1 ≤ a ≤ 9

|C 0 ≤ b, c ≤ 9 (»A n = 123123). J 7 | n.

(3) ' a ∈ Z v 2 - a.

(a) J a² ≡ 1 (mod 8).

(b) 1¬ a² 3 modulo 24 ìXbÝ!õv.

(c) u' 3 - a, J a² ≡ 1 (mod 24).

(4) á gcd(58, 63) = 1, Æá 58 3 modulo 63 ìb¶°D-ô (ÇD3 a ∈ Z¸ÿ 58 × a ≡ 1 (mod 63)). |ìÎbn¶°D-ô®Þ.

(a) ¿à»8t°0 58x + 63y = 1 Ý×àJó. ¬µh0 58 3 modulo 63 ìÝ¶°D-ô.

(b) 0 b ∈ Z v 1 ≤ b ≤ 63, ¸ÿ 58 × b ≡ 47 (mod 63).

———————————– 10 April, 2008

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(5) ' p ≥ 5 Î×Í²ó, J

{−(p − 1)

2 ,−(p − 3)

2 , . . . , −2, −1, 1, 2, . . . ,p − 3 2 ,p − 1

2 } Î×Í reduced residue system modulo p.

(6) |ìÎbn Euler’s Theorem ÝTà.

(a) O 99⁹⁹⁹⁹⁹⁹ t| 26 Ýõó.

(b) ' n ∈ Z v 3 - n. J 9 | n⁷− n¬µhJ n⁷≡ n (mod 63).

(c) ' m, n ∈ N v gcd(m, n) = 1. J m^φ(n)+n^φ(m)≡ 1 (mod mn).

(7) |ìÎbn Fermat’s Little Theorem ÝTà.

(a) J 11 Jt 456⁶⁵⁴+ 123³²¹.

(b) ' p Î×²óv a, b ∈ Z p - a |C p - b. Ju a^p ≡ b^p (mod p)J a ≡ b (mod p), ¬µhJu a^p ≡ b^p (mod p)J a^p ≡ b^p (mod p²).

(c) ' p, q Î8²²óv p − 1 | q − 1. Ju a ∈ Z v gcd(a, pq) = 1, J a^q−1≡ 1 (mod pq).

(8) ¿à Fermat’s Little Theorem 0 11 3 modulo 29 Ý¶°D-ô, ¬ µh 11x ≡ 15 (mod 29).

(9) ' p Î×Í²ó.

(a) Ju a ∈ N (p − 1)/2 < a < p, JD3 b ∈ N 1 ≤ b ≤ (p − 1)/2¸ÿ a ≡ −b (mod p). µh|C Wilson’s Theorem J

(p − 1

2 !)² ≡ (−1)^(p+1)/2 (mod p).

(b) Ju a Îó 1 ≤ a < p−1, JD3ó b 1 < b ≤ p−1

¸ÿ a ≡ −b (mod p). µh|C Wilson’s Theorem J

1²3²5²· · · (p − 4)²(p − 2)² ≡ (−1)^(p+1)/2 (mod p).

(c) ¿àGËÞJ p ≡ 1 (mod 4) ` congruence equation x²≡ −1 (mod p) b.

(10) ' m ∈ N v m > 2. u {r1, r₂, . . . , r_φ(m)} Î×Í modulo m ìÝ reduced residue system,J r1+ r₂+ · · · + r_φ(m)≡ 0 (mod m).

———————————– 17 April, 2008

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Chapter 4. Congruence Equations 7

Chapter 4. Congruence Equations (1)  p ×²ó.

(a) ' f(x) = anxⁿ+· · ·+a₁x+a₀ Í ai ∈ Z. uD3 r1, . . . , r_n+1∈ Z

 f(ri) ≡ 0 (mod p)vE i 6= j /b ri 6≡ r_j (mod p), J

EXb 0 ≤ i ≤ n /b p | ai.

(b) Ê g(x) = (x − 1)(x − 2) · · · (x − (p − 1)) − (x^p−1 − 1). J

g(x) = a_p−2x^p−2+ · · · + a₁x + a₀,ÍEXb 0 ≤ i ≤ p − 2 /b p | ai. (c) ¿à (b) J Wilson’s Theorem.

(2) congruence equation Ý]°ô.ÂÕ9ó94PÝµ.

(a) ' f(x, y) ∈ Z[x, y] | x, y óÝJ;ó94P. Ju m⁰ | m v f(x, y) ≡ 0 (mod m⁰) PJó, J f(x, y) ≡ 0 (mod m) PJó.

(b) J congruence equation 3x²− 7y²≡ 2 (mod 525)PJó.

(3) |ìÝ congruence equation.

(a) O 9x ≡ 21 (mod 30) 3 modulo 30 ìÝXb.

(b) O 18x ≡ 15 (mod 27) 3 modulo 27 ìÝXb.

(4) ×gÝ congruence equation Ý]°ô.ÂÕ9óÝ×g congruence equation.

(a) Ê congruence equation a1x+a₂y ≡ b (mod m).  d = gcd(a1, a₂, m).

Ju d - b, Jh congruence equation P, u d | b, Jh con- gruence equation 3 modulo m ìb dm à.

(b) |ìÝ congruence equation:

(1) 2x + 3y ≡ 4 (mod 7); (2) 3x + 6y ≡ 2 (mod 9).

(c) Þ (a) Ý.ÂÕ n ÍóÝ×g congruence equation.

(5) : (a)









x ≡ 1 (mod 3) x ≡ 2 (mod 4) x ≡ 3 (mod 5) x ≡ 4 (mod 7)

(b)





5x ≡ 3 (mod 7) 2x ≡ 4 (mod 8) 3x ≡ 6 (mod 9)

(c) x² ≡ x (mod 525).

(6) »yõ§ô.ÂÕ!²Ýµ.

(a) J

½ x ≡ b₁ (mod m₁)

x ≡ b₂ (mod m₂) bÐñuv°ugcd(m₁, m₂) | b₁− b₂.

¬JubJÍ3 modulo lcm(m1, m₂)ì°×.

(b) O|ìÐñ: (a)

½ x ≡ 3 (mod 4) x ≡ 1 (mod 6) (b)

½ x ≡ 7 (mod 16) x ≡ 3 (mod 24) (c) Þ (a) Ý.ÂÕ n ÍÐñPÝ.

———————————– 01 May, 2008

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Chapter 5. ÞgÝ Congruence Equations

(1) ¿àg]°|ìÞgÝ congruence equations.

(a) x²+ x ≡ 3 (mod 13) (b) 2x²+ x ≡ 3 (mod 39)

(c) 3x²+ x ≡ 3 (mod 39)

(2) |ìÞgÝ congruence equations.

(a) x²≡ 21 (mod 4) (b) x²≡ 21 (mod 32)

(c) x²≡ 33 (mod 64) (d) x²≡ 40 (mod 64) (e) 2x² ≡ 40 (mod 32) (f) 2x² ≡ 40 (mod 64)

(a) x²≡ 21 (mod 9) (b) x²≡ 18 (mod 27)

(c) x²≡ 31 (mod 81)

(a) x²+ 7x ≡ 15 (mod 216) (b) x²+ 11x ≡ 18 (mod 216)

———————————– 08 May, 2008

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Chapter 5. ÞgÝ Congruence Equations 9

(5) 5½¿à Euler’s Criterion |C Gauss’s Lemma Õ|ì Legendre symbols:

(a) µ11

23

¶

(b) µ−6

11

¶ . (6) ' p Î×²ó, ¿à Gauss’s Lemma J

µ−1 p

¶

= 1 uv°u p ≡ 1 (mod 4).

(7) ' p Î×²ó, a, b ∈ Z v p - a, p - b. J ax² ≡ b (mod p) b

uv°u µa

p

¶ µb p

¶

= 1.

(8) ' p Î×²ó.

(a) ' a, b ∈ Z J a² ≡ b² (mod p)uv°u a ≡ ±b (mod p).

(b) J

µ1 p

¶ +

µ2 p

¶

+ · · · +

µp − 1 p

¶

= 0.

(c) J p ≡ 1 (mod 4) ` µ1

p

¶ +

µ2 p

¶

+ · · · +

µ(p − 1)/2 p

¶

= 0.

(9) |ì&ÆàDJ°J 4k + 1 PÝ²óbPM9Í. ' p1, . . . , p_r ÎXb 4k + 1 PÝ²ó, Ju q Î×²óv q|4p²₁· · · p²_r+ 1,J q ≡ 1 (mod 4) (Ç q ù 4k + 1 P). µhÿë;ÿJ 4k + 1 PÝ²ó bPM9Í.

———————————– 15 May, 2008

(10)

(10) Õ|ìÝ Legendre symbols.

(a)

µ−79 101

¶

(b) µ 91

127

¶

(c)

µ2817 4177

¶ . (11) ' p, q / ²ó.

(a) Ju p = 4q + 1, J x²≡ q (mod p) b.

(b) ' p ≡ q ≡ 3 (mod 4). J x² ≡ p (mod q)buv°u x²≡ −q (mod p) b.

(c) J µ−2

p

¶

= 1 uv°u p ≡ 1, 3 (mod 8).

(d) J µ3

p

¶

= 1 uv°u p ≡ ±1 (mod 12).

(e) 0¸ÿ µ−5

p

¶

= 1 Ý²óÝf.

Chapter 6. Primitive Roots

(1) O|ì order: (a) ord15(8) (b) ord₁₇(9) (2) 13 modulo 15 õ 16 ìP primitive roots.

(3) ' gcd(ordm(a), ord_m(b)) = 1. J ordm(ab) = ord_m(a) · ord_m(b).

(4) Ju ordm(a) = m − 1, J m Ä ²ó.

(5) ' a, n ∈ N v a > 1. J ordaⁿ−1(a) = n,¬µhÿJ n | φ(aⁿ− 1).

———————————– 22 May, 2008

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Chapter 6. Primitive Roots 11

(6) ' m, n ∈ N v gcd(a, mn) = 1.

(a) J lcm(ordm(a), ord_n(a)) | ord_mn(a).

(b) uê' gcd(m, n) = 1, J ordmn(a) | lcm(ord_m(a), ord_n(a)). ¬ µhÿJ gcd(m, n) = 1 ` ordmn(a) = lcm(ord_m(a), ord_n(a)).

(7) á ord101(10) = 4,03 modulo 101 ìXb order 4 Ý-ô¬

1§ã.

(8)  p ×²óv a modulo p ìÝ×Í primitive root.

(a) u b ∈ Z ab ≡ 1 (mod p), J b Î modulo p ìÝ×Í primitive root. µhJ p > 3 `Xb3 modulo p ìÝ primitive root ¶3 modulo p ì congruent to 1.

(b) J Legendre symbol µa

p

¶

= −1.

(c) J

ord_p(−a) =

½ p − 1, u p ≡ 1 (mod 4);

(p − 1)/2, u p ≡ 3 (mod 4).

(9) 0×Jó¸ÍE m ∈ N 3 modulo 13^m õ 2 · 13^m ì/

primitive root.

———————————– 29 May, 2008

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(10) 03 modulo 486 ì primitive root tÝÑJó.

(11) ¾\|ì congruence equation ÎÍb, ub¶ìÍXb.

(a) x³≡ 11 (mod 14).

(b) x⁴≡ 12 (mod 17).

(c) x⁵≡ 13 (mod 18).

(d) x⁶≡ 7 (mod 19).

(12) 3|ì&Þ0XbÝJó a ¸Í congruence equation b.

(a) ax⁹≡ 6 (mod 13).

(b) 7x⁹ ≡ a (mod 13).

(c) ax¹²≡ 9 (mod 17).

(d) 8x¹²≡ a (mod 17).

(13)  p ≥ 5 ²ó.

(a) J x⁴ ≡ −1 (mod p)buv°u p ≡ 1 (mod 8).

(b) ' p ≡ 1 (mod 6) v p - a Ju x³ ≡ a (mod p) b, J3 modulo p ìb 3 Í8².

(c) ' p ≡ 5 (mod 6) J x³ ≡ a (mod p)3 modulo p ìb°×

.

———————————– 05 June, 2008

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Chapter 7. ¯ Diophantine Equations 13

Chapter 7. ¯ Diophantine Equations (1) J|ì Diophantine equation P.

(a) 3x²− 7y² = 2.

(b) x²+ y²+ 1 = 4z.

(2) 0Xb z ≤ 50 Ý primitive Pythagorean triples x, y, z.

(3) ' x, y, z Î×à primitive Pythagorean triple.

(a) J x, y ªb×ÍÎ 3 Ý¹ó.

(b) J x, y ªb×ÍÎ 4 Ý¹ó.

(c) J x, y, z ªb×ÍÎ 5 Ý¹ó.

(d) J 60 | xyz

(4) 0 x²+ 4y² = z² ÝXbÑJó.

(5) J x⁴− y⁴ = z² ^bÑJó.

(6) Ju x, y, z Î×à Pythagorean triple J x, y, z t9Gb×ÍÎJ óÝ¿].

(7) 1|ìJóÎÍ|¶WËÍJóÝ¿]õ, u|Þ¶WËÍ

JóÝ¿]õ.

(a) 207 (b) 637 (c) 522 (d) 605

———————————– 12 June, 2008

a, b, c ∈ Z v c 6= 0, J|ìbnyJtÝÃÍP²

Exercise

a, b, c ∈ Z v c 6= 0, J|ìbnyJtÝÃÍP²