Journal of Integer Sequences, Vol. 1 (1998), Article 98.1.1

On Happy Factorizations

J. H. Conway
Department of Mathematics
Princeton University, Princeton NJ 08544
Email address: conway@math.Princeton.EDU

Abstract: (Supplied by the editors.) It is asserted without proof that every positive integer is the product of a unique "happy couple" of integers. A "happy couple" is an ordered pair of integers of one of three types: (A,A) ; (B,C), with C > 1, where there exist integers R, S such that B R 2 + 1 = C S 2 ; and (D,E) where there exist odd integers T, U such that D T 2 + 2 = E U 2.

Any ordered couple of integers that can be obtained from the positive integers (n,n+d) by dividing them by possibly distinct perfect squares prime to d is called a "d-happy couple", except that couples of the form (m,1) are NOT called 1-happy.

A "happy couple" is just a d-happy couple for d = 0, 1 or 2.

Theorem. Each positive integer N is the product of a unique happy couple.

I call this "the happy factorization" of N, and append a table of happy factorizations, writing a number as

A^2           B.C           or     D:E

according as it is the product of a

0-happy couple (A,A),   1-happy couple (B,C),   or 2-happy couple (D,E). 


                                                                 13^2     
		                                           12^2  1.170
                                                     11^2  1.145 1:171
					       10^2  1.122 2.73  43.4
                                          9^2  1.101 1:123 3.49  1.173
                                     8^2  1.82 2.51  31.4  4.37  29.6
                                7^2  1.65 1:83 103:1 1.125 1.149 7.25
                           6^2  1.50 2.33 3.28 2:52  14.9  6.25  22:8
                      5^2  1.37 1:51 1:67 1.85 5.21  127:1 151:1 59.3
                 4^2  1.26 2.19 4.13 4.17 2.43 1.106 64:2  4:38  2.89
            3^2  1.17 1:27 3.13 1.53 23.3 3:29 1:107 3.43  17.9  1:179
        2^2 1.10 2.9  7.4  2:20 2.27 5.14 4:22 27.4  1.130 7.22  20.9
    1^2 1.5 1:11 1:19 1.29 1.41 11.5 71:1 1.89 1.109 1:131 31.5  1.181
0^2 1.2 2.3 3.4  4.5  5.6  6.7  7.8  8.9  9.10 10.11 11.12 12.13 13.14
1^2 1:3 7:1 1.13 3.7  31:1 1:43 3.19 1.73 13:7 3.37  19.7  1.157 3.61
    2^2 2:4 7.2  2.11 16:2 11.4 1.58 1.74 23.4 7.16  2.67  79.2  23.8
        3^2 3:5  23:1 11.3 5.9  1:59 25:3 3.31 1.113 27:5  3.53  1.185
            4^2  4:6  17.2 23.2 15.4 19.4 47.2 2.57  34:4  80:2  6.31
                 5^2  5:7  47:1 1.61 7.11 19.5 5:23  1.137 7.23  1:374
		      6^2  6:8  31.2 26.3 48:2 4.29  23.6  2.81  47.4
			   7^2  7:9  79:1 1.97 9.13  1:139 1:163 27.7
			        8^2  8:10 49.2 2.59  35.4  4.41  10.19
				     9^2  9:11 1:119 47.3  11.15 191:1
                                          10^2 10:12 71.2  2.83  96:2
                                               11^2  11:13 1:167 1.193
                                                     12^2  12:14 97.2
                                                           13^2  13:15
                                                                 14^2
A HAPPY FACTORIZATION TABLE

I have a truly wonderful proof of the happy factorization theorem, which unfortunately the rest of this page is too small to contain, so I shall have to leave it as an exercise to the reader.

(This is the source for sequences A007966, A007967, A007968, A007969, A007970.)

Received June 28, 1996; published in Journal of Integer Sequences Jan. 1, 1998.

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