Perfect Polynomials. modulo 2. Ugur Caner Cengiz. Lake Forest College. April 7, 2015
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1 Perfect Polynomials modulo 2 Ugur Caner Cengiz Lake Forest College April 7, 2015 Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
2 Outline 1 Definitions modulo 2 What is "Perfect"? Perfect Polynomials Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
3 Outline 1 Definitions modulo 2 What is "Perfect"? Perfect Polynomials 2 Previous Research Others Ours Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
4 Outline 1 Definitions modulo 2 What is "Perfect"? Perfect Polynomials 2 Previous Research Others Ours 3 Our Research The Program Main Results Speed!!! Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
5 Definitions modulo 2 What do you mean by "modulo 2"? In simple terms, a mod b gives the remainder when integer a is divided by non-zero integer b. Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
6 Definitions modulo 2 What do you mean by "modulo 2"? In simple terms, a mod b gives the remainder when integer a is divided by non-zero integer b. Therefore, mod 2 is very simple: if the number is odd, then it is equivalent to 1 and if even, then it is equivalent to 0. Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
7 Definitions modulo 2 What do you mean by "modulo 2"? In simple terms, a mod b gives the remainder when integer a is divided by non-zero integer b. Therefore, mod 2 is very simple: if the number is odd, then it is equivalent to 1 and if even, then it is equivalent to and (mod 2) Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
8 Perfect Numbers Sigma function σ Definitions What is "Perfect"? Definition Lower case Greek letter sigma (σ) symbolizes an arithmetic function that sums the positive divisors of a positive integer. σ(n) = d d n Definition If σ(n) = 2n, then n is perfect. Example Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
9 Perfect Numbers Sigma function σ Definitions What is "Perfect"? Definition Lower case Greek letter sigma (σ) symbolizes an arithmetic function that sums the positive divisors of a positive integer. σ(n) = d d n Definition If σ(n) = 2n, then n is perfect. Example 6 Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
10 Perfect Numbers Sigma function σ Definitions What is "Perfect"? Definition Lower case Greek letter sigma (σ) symbolizes an arithmetic function that sums the positive divisors of a positive integer. σ(n) = d d n Definition If σ(n) = 2n, then n is perfect. Example 6 σ(6) = = 12 Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
11 Perfect Numbers Sigma function σ Definitions What is "Perfect"? Definition Lower case Greek letter sigma (σ) symbolizes an arithmetic function that sums the positive divisors of a positive integer. σ(n) = d d n Definition If σ(n) = 2n, then n is perfect. Example 6 σ(6) = = 12 Hence, 6 is perfect. Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
12 Continuing on σ Definitions What is "Perfect"? Theorem σ is multiplicative over integers. If gcd(m,n) = 1, then σ(mn) = σ(m) σ(n) Example Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
13 Definitions What is "Perfect"? Continuing on σ Theorem σ is multiplicative over integers. If gcd(m,n) = 1, then σ(mn) = σ(m) σ(n) Example 6 = 2 3 where 2 and 3 are prime. Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
14 Definitions What is "Perfect"? Continuing on σ Theorem σ is multiplicative over integers. If gcd(m,n) = 1, then σ(mn) = σ(m) σ(n) Example 6 = 2 3 where 2 and 3 are prime. σ(2)=(2 + 1) and σ(3)=(3+1) Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
15 Definitions What is "Perfect"? Continuing on σ Theorem σ is multiplicative over integers. If gcd(m,n) = 1, then σ(mn) = σ(m) σ(n) Example 6 = 2 3 where 2 and 3 are prime. σ(2)=(2 + 1) and σ(3)=(3+1) σ(2) σ(3)=3 4 = 12 = σ(6). Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
16 Definitions What is "Perfect"? Continuing on σ Theorem σ is multiplicative over integers. If gcd(m,n) = 1, then σ(mn) = σ(m) σ(n) Example 6 = 2 3 where 2 and 3 are prime. σ(2)=(2 + 1) and σ(3)=(3+1) σ(2) σ(3)=3 4 = 12 = σ(6). 728 = Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
17 Definitions What is "Perfect"? Continuing on σ Theorem σ is multiplicative over integers. If gcd(m,n) = 1, then σ(mn) = σ(m) σ(n) Example 6 = 2 3 where 2 and 3 are prime. σ(2)=(2 + 1) and σ(3)=(3+1) σ(2) σ(3)=3 4 = 12 = σ(6). 728 = σ(728) = = 1680 Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
18 Definitions What is "Perfect"? Continuing on σ Theorem σ is multiplicative over integers. If gcd(m,n) = 1, then σ(mn) = σ(m) σ(n) Example 6 = 2 3 where 2 and 3 are prime. σ(2)=(2 + 1) and σ(3)=(3+1) σ(2) σ(3)=3 4 = 12 = σ(6). 728 = σ(728) = = 1680 Note that σ(p q )= (p q + p q p + 1) Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
19 Definitions What is "Perfect"? Continuing on σ Theorem σ is multiplicative over integers. If gcd(m,n) = 1, then σ(mn) = σ(m) σ(n) Example 6 = 2 3 where 2 and 3 are prime. σ(2)=(2 + 1) and σ(3)=(3+1) σ(2) σ(3)=3 4 = 12 = σ(6). 728 = σ(728) = = 1680 Note that σ(p q )= (p q + p q p + 1) So, σ(2 3 ) σ(7) σ(13) = ( )(7 + 1)(13 + 1) Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
20 Definitions What is "Perfect"? Continuing on σ Theorem σ is multiplicative over integers. If gcd(m,n) = 1, then σ(mn) = σ(m) σ(n) Example 6 = 2 3 where 2 and 3 are prime. σ(2)=(2 + 1) and σ(3)=(3+1) σ(2) σ(3)=3 4 = 12 = σ(6). 728 = σ(728) = = 1680 Note that σ(p q )= (p q + p q p + 1) So, σ(2 3 ) σ(7) σ(13) = ( )(7 + 1)(13 + 1) σ(2 3 ) σ(7) σ(13) = = 1680 Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
21 Polynomials mod 2 Definitions Perfect Polynomials For a polynomial mod 2, the coefficients are mod 2. Thus, 5x 7 +9x 6 +16x 5 +48x 4 +x 3 +4x 2 +71x +1 x 7 +x 6 +x 3 +x +1 Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
22 Polynomials mod 2 Definitions Perfect Polynomials For a polynomial mod 2, the coefficients are mod 2. Thus, 5x 7 +9x 6 +16x 5 +48x 4 +x 3 +4x 2 +71x +1 x 7 +x 6 +x 3 +x +1 x = 0 has no real roots. It s irreducible. Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
23 Polynomials mod 2 Definitions Perfect Polynomials For a polynomial mod 2, the coefficients are mod 2. Thus, 5x 7 +9x 6 +16x 5 +48x 4 +x 3 +4x 2 +71x +1 x 7 +x 6 +x 3 +x +1 x = 0 has no real roots. It s irreducible. Consider (x + 1) 2 modulo 2 Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
24 Polynomials mod 2 Definitions Perfect Polynomials For a polynomial mod 2, the coefficients are mod 2. Thus, 5x 7 +9x 6 +16x 5 +48x 4 +x 3 +4x 2 +71x +1 x 7 +x 6 +x 3 +x +1 x = 0 has no real roots. It s irreducible. Consider (x + 1) 2 modulo 2 (x + 1) 2 = (x + 1) (x + 1) = x 2 + 2x Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
25 Polynomials mod 2 Definitions Perfect Polynomials For a polynomial mod 2, the coefficients are mod 2. Thus, 5x 7 +9x 6 +16x 5 +48x 4 +x 3 +4x 2 +71x +1 x 7 +x 6 +x 3 +x +1 x = 0 has no real roots. It s irreducible. Consider (x + 1) 2 modulo 2 (x + 1) 2 = (x + 1) (x + 1) = x 2 + 2x x 2 + 2x x (mod 2) Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
26 Definitions Perfect Polynomials σ on polynomials Definition If σ(a) = A, then A is a perfect polynomial. Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
27 Definitions Perfect Polynomials σ on polynomials Definition If σ(a) = A, then A is a perfect polynomial. For example, x 2 + x = x (x + 1) Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
28 Definitions Perfect Polynomials σ on polynomials Definition If σ(a) = A, then A is a perfect polynomial. For example, x 2 + x = x (x + 1) σ(x 2 + x) = (x 2 + x) + (x + 1) + x + 1 = x 2 + 3x + 2 Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
29 Definitions Perfect Polynomials σ on polynomials Definition If σ(a) = A, then A is a perfect polynomial. For example, x 2 + x = x (x + 1) σ(x 2 + x) = (x 2 + x) + (x + 1) + x + 1 = x 2 + 3x + 2 x 2 + 3x + 2 x 2 + x (mod 2) Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
30 Definitions Perfect Polynomials σ on polynomials Definition If σ(a) = A, then A is a perfect polynomial. For example, x 2 + x = x (x + 1) σ(x 2 + x) = (x 2 + x) + (x + 1) + x + 1 = x 2 + 3x + 2 x 2 + 3x + 2 x 2 + x (mod 2) So σ(x 2 + x) = x 2 + x (mod 2) x 2 + x is a perfect polynomial mod 2. Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
31 Previous Research Others Canaday, Gallardo and Rahavandrainy E. F. Canaday Gallardo and Rahavandrainy Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
32 Previous Research Others Canaday, Gallardo and Rahavandrainy E. F. Canaday Perfect polynomials mod 2 exist in two ways: x h (x + 1) k A and B 2, where B is relatively prime to x(x + 1) Gallardo and Rahavandrainy Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
33 Previous Research Others Canaday, Gallardo and Rahavandrainy E. F. Canaday Perfect polynomials mod 2 exist in two ways: x h (x + 1) k A and B 2, where B is relatively prime to x(x + 1) Also, he found an infinite class of perfects: x 2n 1 (x + 1) 2n 1 Gallardo and Rahavandrainy Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
34 Previous Research Others Canaday, Gallardo and Rahavandrainy E. F. Canaday Perfect polynomials mod 2 exist in two ways: x h (x + 1) k A and B 2, where B is relatively prime to x(x + 1) Also, he found an infinite class of perfects: x 2n 1 (x + 1) 2n 1 Conjecture that every perfect is divisible by x(x + 1) In other words, no odd perfects Gallardo and Rahavandrainy Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
35 Previous Research Others Canaday, Gallardo and Rahavandrainy E. F. Canaday Perfect polynomials mod 2 exist in two ways: x h (x + 1) k A and B 2, where B is relatively prime to x(x + 1) Also, he found an infinite class of perfects: x 2n 1 (x + 1) 2n 1 Conjecture that every perfect is divisible by x(x + 1) In other words, no odd perfects Gallardo and Rahavandrainy Proved that odd perfects have at least 5 distinct irreducible factors. Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
36 Previous Research Others Figure: Canaday s list for perfects Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
37 Previous Research The Algorithm to Find the Perfect Polynomials Ours Check if σb = B. Output B. Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
38 Previous Research Ours The Algorithm to Find the Perfect Polynomials Check if σb = B. Output B. If not, compute D where D = σ(b) / gcd(b, σ(b)) Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
39 Previous Research Ours The Algorithm to Find the Perfect Polynomials Check if σb = B. Output B. If not, compute D where D = σ(b) / gcd(b, σ(b)) If gcd(b, D) > 1, then stop. No output! Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
40 Previous Research Ours The Algorithm to Find the Perfect Polynomials Check if σb = B. Output B. If not, compute D where D = σ(b) / gcd(b, σ(b)) If gcd(b, D) > 1, then stop. No output! If the polynomial passes the test on step 3, then let P be the greatest factor of D. Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
41 Previous Research Ours The Algorithm to Find the Perfect Polynomials Check if σb = B. Output B. If not, compute D where D = σ(b) / gcd(b, σ(b)) If gcd(b, D) > 1, then stop. No output! If the polynomial passes the test on step 3, then let P be the greatest factor of D. Restart the algorithm taking BP, BP 2, BP 3,..., BP k where degree of BP k < K. Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
42 Previous Research Ours Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
43 Our Research Beginning steps - primperf() The Program Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
44 Our Research Main Results Results up to degree 200 x (x + 1) 2 (x 2 + x + 1) x (x + 1) 2 (x 2 + x + 1) 2 (x 4 + x + 1) (x + 1) x 2 (x 2 + x + 1) (x + 1) x 2 (x 2 + x + 1) 2 (x 4 + x + 1) x 3 (x + 1) 4 (x 4 + x 3 + 1) x 3 (x + 1) 6 (x 3 + x + 1) (x 3 + x 2 + 1) (x + 1) 3 x 4 (x 4 + x 3 + x 2 + x + 1) x 4 (x + 1) 4 (x 4 + x 3 + 1) (x 4 + x 3 + x 2 + x + 1) x 4 (x + 1) 6 (x 3 + x + 1) (x 3 + x 2 + 1) (x 4 + x 3 + x 2 + x + 1) (x + 1) 3 x 6 (x 3 + x + 1) (x 3 + x 2 + 1) (x + 1) 4 x 6 (x 3 + x + 1) (x 3 + x 2 + 1) (x 4 + x 3 + 1) x (x + 1) x 3 (x + 1) 3 x 7 (x + 1) 7 x 15 (x + 1) 15 x 31 (x + 1) 31 and x 63 (x + 1) 63 Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
45 Speed Our Research Speed!!! def sigma1(x, y): return (x y+1 1)/(x 1) def sigma2(x, y): sum = 0 for pow in range(0, y+1): sum = sum + (x pow ) return sum sigma1 and sigma2 speed testing Dynamic Programming Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
46 Our Research Speed!!! Figure: FAST! Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
47 Summary Summary A perfect polynomial equals the sum of its divisors. As Canaday thought there are no odd perfect polynomials up to degree 200. My program is relatively fast and finds the perfect polynomials. Future Plans To check higher degrees Show odd perfect polynomials mod 2 have at least 6 factors Work on a paper Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
48 For Further Information Summary E.F. Canaday The Sum of The Divisors of a Polynomial. Duke Mathematical Journal, 8(4): , 1941 L. Gallardo. and O. Rahavandrainy. Odd Perfect Polynomials over F 2 Journal de Théeorie des Nombres de Bordeaux, 19(1): , L. Gallardo. and O. Rahavandrainy. There is no odd perfect polynomial over F 2 with four prime factors Portugaliae Mathematica, 66(2): , Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
49 Summary Thank You! (Any Questions?) Ugur Caner Cengiz Perfect Polynomials Student Symposium / 18
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