Perfect Number Solution Key

	n:=1	n:=2	n:=3	n:=4	n:=5	n:=6	n:=7	n:=8	n:=9
a(n)	32	512	8192	131072	2097152	33554432	536870912	8589934592	137438953472
b(n)	28	496	8128	130816	2096128	33550336	536854528	8589869056	137438691328
c(n)	4	16	64	256	1024	4096	16384	65536	262144
f(n)	7	31	127	511	2047	8191	32767	131071	524287
g(n)	3	5	7	9	11	13	15	17	19

Prime numbers (cyan background). Perfect numbers (yellow background).

The product sequence algorithm of c(n) is the solution key to unlocking all Prime numbers (shaded cyan) and Perfect numbers (shaded yellow) for numbers greater than 6 in our number system. The idea for the algorithm comes from my article: "Prime Consideration for the Perfect Number" shown below.

The ‘n’ sequence (n:=1, n:=2, .. , n:=9) can be easily increased and iterated in a software program. Results for the ‘n’ sequence, shown in the chart above, were derived from MathCad Professional and Mathematica computer programs.

If f(n) is prime, then b(n) is perfect. Also, if f(n) is prime, it will be a Mersenne prime. There are various ways to check a number for primeness. Factoring f(n) can be very slow on extremely large numbers. Using the IsPrime function (Mathcad) or PrimeQ function (Mathematica) takes much less time but gives a probability that a number is Prime. Use of IsPrime or PrimeQ may need to be verified on very large numbers by an independent source. Check out GIMPS (The Great Internet Mersenne Prime Search) at http://www.mersenne.org/prime.htm.

For an excellent reference on searching for Primes using various algorithms, read Prime Numbers, A Computational Perspective, by Richard Crandall and Carl Pomerance, Springer-Verlag New York, Inc, 2001. This book contains many old and new algorithms that were used in finding extremely large Mersenne Primes such as 2^13,466,917 –1.

 

Prime Consideration for the Perfect Number

A	8	32	512	8192	131072	2097152	33554432	536870912	8589934592	137438953472
B	6	28	496	8128	130816	2096128	33550336	536854528	8589869056	137438691328
C	2	4	16	64	256	1024	4096	16384	65536	262144
D	2	22	42	82	162	322	642	1282	2562	5122
E	21	22	24	26	28	210	212	214	216	218
F	3	7	31	127	511	2047	8191	32767	131071	524287
G	2	3	5	7	9	11	13	15	17	19

Prime numbers (cyan background). Perfect numbers (yellow background).

A = Computer-type numbers. Starting with 3^rd numbered column from the left, multiply previous number by 16. Repeat for additional numbers.

Examples: 32*16 = 512, 512*16 = 8192, 8192*16 =131072, …,

B = Known Perfect numbers in red with yellow background using Euclid’s theorem: [B = 2^G-1(2^G-1)]. B = C*F, B = D*F, B = E*F

Examples: 2^2-1(2²-1) = 6, 2^3-1(2³-1) = 28, 2^5-1(2⁵-1) = 496, …,

(Notice that the numbers in row B can be found by subtracting row C from row A.)

C = Subtract row B from row A. (Notice computer-type numbers again). C = 2^G-1

Examples: 8-6 = 2, 32-28 = 4, 512-496 = 16, …,

Using 2^G-1 = C, Examples: 2^2-1 = 2, 2^3-1 = 4, 2^5-1 = 16, …,

D = Row C converted to (n*2)² beginning with 3^rd numbered column from the left where ‘n’ equals the base number of the previous column.

Examples: (2*2)² = 16, (4*2) ² = 64, (8*2) ² = 256, …,

E = Row C converted to 2ⁿ⁺² beginning with 3^rd numbered column from the left where ‘n’ equals the exponent of previous column.

Examples: 2²⁺² = 2⁴ = 16, 2⁴⁺² = 2⁶ = 64, 2⁶⁺² = 2⁸ = 256, …,

F = First odd number found by factoring row B. (Notice all Primes in the chart have a cyan background). F = 2^G-1, F = 2*C-1

Starting with 3^rd numbered column from the left, apply 4(n+1)-1 where ‘n’ equals the number of the previous column.

Examples: 4(7+1)-1 = 31, 4(31+1)-1 = 127, 4(127+1)-1 = 511, …,

Using 2^G-1 = F, Examples: 2²-1 = 3, 2³-1 = 7, 2⁵-1 = 31, …,

G = Base Prime numbers. These numbers also represent the total number of Binary 1s in each number from rows B and F.

Examples from row B: 6 = 110, 28 = 11100, 496 = 111110000, 8128 = 1111111000000, …,

(Notice the number of 0s in each binary number equals the corresponding exponent in row E.)

Examples from row F: 3 = 11, 7 = 111, 31 = 11111, 127 = 1111111, …,

If row F, and/or rows F and G (not row G only) are Primes, there will be a Perfect number in row B.

(Notice number 11 in row G is Prime but 2047 in row F is not, therefore, no associated Perfect number exists in row B.)

Also, rows F and G can be used to search for double-Mersenne Primes (MM_p) using MM_G = 2^F-1

Examples: MM₂ = 2³-1 = 7, MM₃ = 2⁷-1 = 127, MM₅ = 2³¹-1 = 2147483647, MM₇ = 2¹²⁷-1 = 170141183460469231731687303715884105727

(Notice that MM_2, MM_3, MM_5, MM₇ all equal primes.)

Does this sequence hold true for the remaining double-Mersenne Primes? Keep in mind that row F reflects the Mersenne Primes that can be used to equate to a Perfect number. There are many sequential patterns in the above chart that can be used in solving the next Perfect or Prime (Mersenne or double-Mersenne) number.

Continue the sequences indefinitely in the chart to reveal all Perfect and Prime numbers. Instead of putting every positive integer into Euclid’s theorem: 2^n-1(2ⁿ-1) to find the next Perfect number, consider using the concepts from the chart to save at least half the time necessary to solve for the next Perfect number.

By Dan Thomasson – November 14, 2001, Dthomasson@Carolina.rr.com

Copyright(C) 2001, Dan Thomasson. This information is copyrighted but may be copied or reproduced in any fashion provided the copyright notice accompanies the charts and analysis.

www.BordersChess.org/Perfect-Key.htm   modified 2006.12.14