Method for efficiently factoring large semiprime numbers.

1. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
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The Babylonian way of factoring RSA-129
By M.W.R. Karskens
The paper describes the method for efficiently factoring RSA-Numbers in the sexagesimal number 60 by application to RSA-129. The method is derived from
the book “On the Theory of Numbers, Prime Numbers and Enlightenment,” Chapter 3 Babylonian Number Theory, The Hague: Sexagesimal Publishing, 2017
pp. 43-45.
Step 1:
Convert RSA-129 to RSA-129 base 60.
RSA-129
114381625757888867669235779976146612010218296721242362562561842935706935245733897830597123563958705058989075147599290026879
543541
129 decimal places
RSA-129b60
001.04.30.30.53.27.55.43.10.03.20.24.52.53.06.50.47.50.58.59.05.27.29.47.31.04.05.36.57.51.05.41.39.42.45.43.26.02.30.52.46.02.23.49.09.09.59.46.14.
52.33.54.05.58.23.42.46.29.41.37.48.55.25.30.05.42.37.50.46.52.42.05.41
073.72.71.70.69.68.67.66.65.64.63.62.61.60.59.58.57.56.55.54.53.52.51.50.49.48.47.46.45.44.43.42.41.40.39.38.37.36.35.34.33.32.31.30.29.28.27.26.25.
24.23.22.21.20.19.18.17.16.15.14.13.12.11.10.09.08.07.06.05.04.03.02.01
73 sexagesimal places
RSA-129base60 is 73 sexagesimal places.
The last sexagesimal place is “41”.

2. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
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Step 2:
From the Standard Sexagesimal Prime Multiplication Table:
Note the tables for .41:
The Standard Sexagesimal Prime Multiplication Table (p x q = n, p>5)
Last place
.07 1.01 x 07 = 7.07 13 x 19 = 4.07 31 x 37 = 19.07 43 x 1.49 = 1.18.07 11 x 17 = 3.07 23 x 29 = 11.07 41 x 47 = 32.07 53 x 59 = 52.07
.13 1.01 x 13 = 13.13 07 x 19 = 2.13 31 x 43 = 22.13 37 x 1.49 = 1.07.13 11 x 23 = 4.13 17 x 29 = 8.13 41 x 53 = 36.13 47 x 59 = 46.13
.19 1.01 x 19 = 19.19 07 x 37 = 4.19 13 x 43 = 9.19 31 x 1.49 = 56.19 11 x 29 = 5.19 17 x 47 = 13.19 23 x 53 = 20.19 41 x 59 = 40.19
.31 1.01 x 31 = 31.31 07 x 13 = 1.31 19 x 1.49 = 34.31 37 x 43 = 26.31 11 x 41 = 7.31 17 x 23 = 6.31 29 x 59 = 28.31 47 x 53 = 41.31
.37 1.01 x 37 = 37.37 07 x 31 = 3.37 13 x 1.49 = 23.37 19 x 43 = 13.37 11 x 47 = 8.37 17 x 41 = 11.37 23 x 59 = 22.37 29 x 53 = 25.37
.43 1.01 x 43 = 43.43 07 x 1.49 12.43 13 x 31 = 6.43 19 x 37 = 11.43 11 x 53 = 9.43 17 x 59 = 16.43 23 x 41 = 15.43 29 x 47 = 22.43
.49 1.01 x 1.49 = 1.50.49 7 x 7 = 49 13 x 13 = 2.49 19 x 31 = 9.49 37 x 37 = 22.49 43 x 43 = 30.49 11 x 59 = 10.49 17 x 17 = 4.49 23 x 23 = 8.49 29 x 41 = 19.49 47 x 47 = 36.49 53 x 53 = 46.49
1.01 1.01 x 1.01 = 1.02.01 07 x 43 = 5.01 13 x 37 = 8.01 19 x 19 = 6.01 31 x 31 = 16.01 1.49 x 1.49 = 3.18.01 11 x 11 = 2.01 17 x 53 = 15.01 23 x 47 = 18.01 29 x 29 = 14.01 41 x 41 = 28.01 59 x 59 = 58.01
.11 1.01 x 11 = 11.11 07 x 53 = 6.11 13 x 47 = 10.11 19 x 29 = 9.11 31 x 31 = 16.01 37 x 23 = 14.11 43 x 17 = 12.11 1.49 x 59 = 1.47.11
.17 1.01 x 17 = 17.17 07 x 11 = 1.17 13 x 29 = 6.17 19 x 23 = 7.17 31 x 47 = 24.17 37 x 41 = 25.17 43 x 59 = 42.17 1.49 x 53 = 1.36.17
.23 1.01 x 23 = 23.23 07 x 29 = 3.23 13 x 11 = 2.23 19 x 17 = 5.23 31 x 53 = 27.23 37 x 59 = 36.23 43 x 41 = 29.23 1.49 x 47 = 1.25.23
.29 1.01 x 29 = 29.29 07 x 47 = 5.29 13 x 53 = 11.29 19 x 11 = 3.29 31 x 59 = 30.29 37 x 17 = 10.29 43 x 23 = 16.29 1.49 x 41 = 1.14.29
.41 1.01 x 41 = 41.41 07 x 23 = 2.41 13 x 17 = 3.41 19 x 59 = 18.41 31 x 11 = 5.41 37 x 53 = 32.41 43 x 47 = 33.41 1.49 x 29 = 52.41
.47 1.01 x 47 = 47.47 07 x 41 = 4.47 13 x 59 = 12.47 19 x 53 = 16.47 31 x 17 = 8.47 37 x 11 = 6.47 43 x 29 = 20.47 1.49 x 23 = 41.47
.53 1.01 x 53 = 53.53 07 x 59 = 6.53 13 x 41 = 8.53 19 x 47 = 14.53 31 x 23 = 11.53 37 x 29 = 17.53 43 x 11 = 7.53 1.49 x 17 = 30.53
.59 1.01 x 59 = 59.59 07 x 17 = 1.59 13 x 23 = 4.59 19 x 41 = 12.59 31 x 29 = 14.59 37 x 47 = 28.59 43 x 53 = 37.59 1.49 x 11 = 19.59
.41 1.01x 41= 41.41 07x 23= 2.41 13x 17= 3.41 19x 59= 18.41 31x 11= 5.41 37x 53= 32.41 43x 47= 33.41 1.49x 29= 52.41

3. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
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Step 3:
Construct the 8 standard Multiplication Tables p x q from each of the 8 entries p x q = .41
Increase p and q in each table with 605
sexagesimal places, i.e. each table contains (605
)2
product numbers.
The 8 tables below show the specific tables extended an hour by an hour (602
)2
1.
.07 x .23 =
.41 Difference ↓ 23.00 1.23.00 2.23.00 3.23.00 1.00.23.00 1.00.00.23.00
07 x 23 .23 1.23 2.23 3.23 .. 1.00.23 .. 1.00.00.23 ..
→ 7.00 .07 2.41 9.41 16.41 23.41 7.02.41 7.00.02.41
1.07.00 1.07 25.41 1.32.41 2.39.41 3.46.41
2.07.00 2.07 48.41 2.55.41 5.02.41 7.09.41
3.07.00 3.07 1.11.41 4.18.41 7.25.41 10.32.41
4.07.00 4.07 1.34.41 5.41.41 9.48.41 13.55.41
..
1.00.07.00 1.00.07 23.02.41 1.00.30.02.41 1.00.07.23.02.41
..
1.00.00.07.00 1.00.00.07 23.0.02.41 1.00.23.07.02.41 1.00.00.30.00.02.41
..

4. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
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2.
.13 x .17 =
.41 Difference 17.00 1.17.00 2.17.00 3.17.00 1.00.17.00 1.00.00.17.00
13 x 17 .17 1.17 2.17 3.17 .. 1.00.17 .. 1.00.00.17 ..
13.00 13 3.41 16.41 29.41 42.41 13.03.41 13.00.03.41
1.13.00 1.13 20.41 1.33.41 2.46.41 3.59.41
2.13.00 2.13 37.41 2.50.41 5.03.41 7.16.41
3.13.00 3.13 54.41 4.07.41 7.20.41 10.33.41
4.13.00 4.13 1.11.41 5.24.41 9.37.41 13.50.41
..
1.00.13.00 1.00.13. 17.03.41 1.00.30.03.41 1.00.13.17.03.41
..
1.00.00.13.00 1.00.00.13 17.00.03.41 1.00.17.13.03.41 1.00.00.30.00.03.41
..

5. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
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3.
.31 x .11 =
.41 Difference 11.00 1.11.00 2.11.00 3.11.00 .. 1.00.11.00 1.00.00.11.00
31 x 11 .11 1.11 2.11 3.11 .. 1.00.11 .. 1.00.00.11 ..
31.00 .31 5.41 36.41 1.07.41 1.38.41 31.05.41 31.00.05.41
1.31.00 1.31 16.41 1.47.41 3.18.41 4.49.41
2.31.00 2.31 27.41 2.58.41 5.29.41 8.00.41
3.31.00 3.31 38.41 4.09.41 7.40.41 11.11.41
4.31.00 4.31 49.41 5.20.41 9.51.41 14.22.41
..
1.00.31.00 1.00.31 11.05.41 1.00.42.05.41 1.00.31.11.05.41
..
1.00.00.31.00 1.00.00.31 11.0.05.41 1.00.11.31.05.41 1.00.00.42.00.05.41
..

6. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
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4.
.19 x .59 =
.41 Difference 59.00 1.59.00 2.59.00 3.59.00 .. 1.00.59.00 1.00.00.59.00
19 x 59 .59 1.59 2.59 3.59 .. 1.00.59 .. 1.00.00.59 ..
19.00 .19 18.41 37.41 56.41 1.15.41 19.18.41 19.00.18.41
1.19.00 1.19 1.17.41 2.36.41 3.55.41 5.14.41
2.19.00 2.19 2.16.41 4.35.41 6.54.41 9.13.41
3.19.00 3.19 3.15.41 6.34.41 9.53.41 13.12.41
4.19.00 4.19 4.14.41 8.33.41 12.52.41 17.11.41
..
1.00.19.00 1.00.19 59.18.41 1.01.18.18.41 1.00.19.59.18.41
..
1.00.00.19.00 1.00.00.19 59.00.18.41 1.00.59.19.18.41 1.00.01.18.00.18.41
..

7. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
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5.
.37 x .53 =
.41 Difference ↓ 53.00 1.53.00 2.53.00 3.53.00 1.00.53.00 1.00.00.53.00
37 x 53 .53 1.53 2.53 3.53 .. 1.00.53 .. 1.00.00.53 ..
→ 37.00 .37 32.41 1.09.41 1.46.41 2.23.41 37.32.41 37.00.32.41
1.37.00 1.37 1.25.41 3.02.41 4.39.41 6.16.41
2.37.00 2.37 2.18.41 4.55.41 7.32.41 10.09.41
3.37.00 3.37 3.11.41 6.48.41 10.25.41 14.02.41
4.37.00 4.37 4.04.41 8.41.41 13.18.41 17.55.41
..
1.00.37.00 1.00.37 53.32.41 1.01.30.32.41 1.00.37.53.32.41
..
1.00.00.37.00 1.00.00.37 53.00.32.41 1.00.53.37.32.41 1.00.01.30.00.32.41
..

8. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
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6.
.43 x .47 =
.41 Difference 47.00 1.47.00 2.47.00 3.47.00 .. 1.00.47.00 1.00.00.47.00
43 x 47 .47 1.47 2.47 3.47 .. 1.00.47 .. 1.00.00.47 ..
43.00 .43 33.41 1.16.41 1.59.41 2.42.41 43.33.41 43.0.33.41
1.43.00 1.43 1.20.41 3.03.41 4.46.41 6.29.41
2.43.00 2.43 2.07.41 4.50.41 7.33.41 10.16.41
3.43.00 3.43 2.54.41 6.37.41 10.20.41 14.03.41
4.43.00 4.43 3.41.41 8.24.41 13.07.41 17.50.41
..
1.00.43.00 1.00.43 47.33.41 1.01.30.33.41 1.00.43.47.33.41
..
1.00.00.43.00 1.00.00.43 47.00.33.41 1.00.47.43.33.41 1.00.01.30.00.33.41
..

9. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
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7.
1.01 x .41 =
.41 Difference 41.00 1.41.00 2.41.00 3.41.00 1.00.41.00 1.00.00.41.00
29 x 41 .41 1.41 2.41 3.41 .. 1.00.41 .. 1.00.00.41 ..
1.01.00 1.01 41.41 1.42.41 2.43.41 3.44.41 1.01.41.41 1.01.00.41.41
2.01.00 2.01 1.22.41 3.23.41 5.24.41 7.25.41
3.01.00 3.01 2.03.41 5.04.41 8.05.41 11.06.41
4.01.00 4.01 2.44.41 6.45.41 10.46.41 14.47.41
..
1.00.01.00 1.00.01 41.0.41 1.00.42.00.41 1.00.01.41.00.41
..
1.00.00.01.00 1.00.00.01 41.00.00.41 1.00.41.01.00.41 1.00.00.42.00.00.41
..

10. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
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8.
1.49 x .29
Difference 29.00 1.29.00 2.29.00 3.29.00 .. 1.00.29.00 1.00.00.29.00
1.49 x 29 .29 1.29 2.29 3.29 .. 1.00.29 .. 1.00.00.29 ..
1.49.00 1.49 52.41 2.41.41 4.30.41 6.19.41 1.49.52.41 1.49.00.52.41
2.49.00 2.49 1.21.41 4.10.41 6.59.41 9.48.41
3.49.00 3.49 1.50.41 5.39.41 9.28.41 13.17.41
4.49.00 4.49 2.19.41 7.08.41 11.57.41 16.46.41 4.51.19.41 4.49.2.19.41
..
1.00.49.00 1.00.49 29.23.41 1.01.18.23.41 1.00.49.29.23.41
..
1.00.00.49.00 1.00.00.49 29.00.23.41 1.00.29.49.23.41 1.00.01.18.00.23.41
..

11. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
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Step 4:
Sieve the 8 tables for a match on the last 6 sexagesimal places of RSA-129:
50.46.52.42.05.41 .
An exact match in the .41= -.53 x -.37 Multiplication Table.
At p6 = 40.11.35.46.48.53
and q6 = 01.05.57.30.09.37
we find 44.11.04.58.56.50.46.52.42.05.41
.41 ↓ 53.00 46.48.53.00 35.46.48.53.00 1.00.00.00.53.00 11.35.46.48.53.00 40.11.35.46.48.53.00
p x q = .41 .53 .. 46.48.53 .. 35.46.48.53 .. 1.00.00.00.53 .. 11.35.46.48.53 .. p6 40.11.35.46.48.53
→ 37.00 37 32.41
..
30.09.37.00 30.09.37 23.31.56.42.05.41
..
1.00.00.37.00 1.00.00.37
..
57.30.09.37.00 57.30.09.37 34.17.27.34.52.42.05.41
..
1.00.00.00.37.00 1.00.00.00.37
..
5.57.30.09.37.00 5.57.30.09.37 1.09.05.43.17.46.52.42.05.41
..
1.00.00.00.00.37.00 1.00.00.00.00.37
..
1.05.57.30.09.37.00 q6 1.05.57.30.09.37 44.11.04.58.56.50.46.52.42.05.41
..
1.00.00.00.30.09.37.00 1.00.00.00.00.00.37

12. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
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Step 5:
Make a dedicated multiplication table for
p6 = 40.11.35.46.48.53
q6 = 01.05.57.30.09.37
p6 x q6 = 44.11.04.58.56.50.46.52.42.05.41
Sieve the table for a match on the last sexagesimal 9 places of RSA-129:
05.42.37.50.46.52.42.05.41
09.08.07.06.05.04.03.02.01 - sexagesimal places
At p9 = 46.20.24.40.11.35.46.48.53
and q9 = 19.28.01.01.05.57.30.09.37 .
We find p9 x q9 = 15.02.06.07.25.46.56.07.32.05.42.37.50.46.52.42.05.41 .
.37 x .53 = .41
p6 x q6 = 40.11.35.46.48.53 x 01.05.57.30.09.37 = ↓ 01.05.57.30.09.37.00.00.00.00.00.00 .09.37.00.00.00.00.00.00 .09.37.00.00.00.00.00.00 0.09.37.00.00.00.00.00.00 .57.30.09.37.00.00.00.00.00.00
01.05.57.30.09.37 1.01.05.57.30.09.37 .. 1.00.01.05.57.30.09.37 .. 1.00.00.01.05.57.30.09.37 .. 1.00.00.00.01.05.57.30.09.37
→ 40.11.35.46.48.53.00.00.00.00.00.00. 40.11.35.46.48.53 44.11.04.58.56.50.46.52.42.05.41
1.40.11.35.46.48.53.00.00.00.00.00.00. 1.40.11.35.46.48.53
2.40.11.35.46.48.53.00.00.00.00.00.00. 2.40.11.35.46.48.53
..
1.00.40.11.35.46.48.53.00.00.00.00.00.00. 1.00.40.11.35.46.48.53
..
1.00.00.40.11.35.46.48.53.00.00.00.00.00.00. 1.00.00.40.11.35.46.48.53
..
1.00.00.00.40.11.35.46.48.53.00.00.00.00.00.00. 1.00.00.00.40.11.35.46.48.53
..
..

13. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
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Step 6:
Make a dedicated multiplication table for
p9 = 46.20.24.40.11.35.46.48.53 ,
q9 = 19.28.01.01.05.57.30.09.37
p9 x q9 = 15.02.06.07.25.46.56.07.32.05.42.37.50.46.52.42.05.41
, and extend the table by three sexagesimal places.
Sieve the table for a match on the last sexagesimal 12 places of RSA-129:
55.25.30.05.42.37.50.46.52.42.05.41
12.11.10.09.08.07.06.05.04.03.02.01 - sexagesimal places
At p12 = 18.34.18.46.20.24.40.11.35.46.48.53
and q12 = 49.12.44.19.28.01.01.05.57.30.09.37 .
We find p12 x q12 = 15.13.57.54.47.14.29.04.25.05.44.31.55.25.30.05.42.37.50.46.52.42.05.41

14. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
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Step 7:
Make a dedicated multiplication table for
p12 = 18.34.18.46.20.24.40.11.35.46.48.53
q12 = 49.12.44.19.28.01.01.05.57.30.09.37 .
p12 x q12 = 15.13.57.54.47.14.29.04.25.05.44.31.55.25.30.05.42.37.50.46.52.42.05.41 .
Sieve the table for a match on the last 15 sexagesimal places of RSA-129:
41.37.48.55.25.30.05.42.37.50.46.52.42.05.41
15.14.13.12.11.10.09.08.07.06.05.04.03.02.01 - sexagesimal places
At p15 = 04.48.55.18.34.18.46.20.24.40.11.35.46.48.53
and q15 = 57.08.14.49.12.44.19.28.01.01.05.57.30.09.37
we find 4.35.08.15.22.57.15.11.59.12.35.35.00.29.55.41.37.48.55.25.30.05.42.37.50.46.52.42.05.41 .

15. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
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Step 8:
Make a dedicated multiplication table for
p15 = 04.48.55.18.34.18.46.20.24.40.11.35.46.48.53
q15 = 57.08.14.49.12.44.19.28.01.01.05.57.30.09.37
p15 x q15 = 4.35.08.15.22.57.15.11.59.12.35.35.00.29.55.41.37.48.55.25.30.05.42.37.50.46.52.42.05.41
Sieve the table for a match on the last sexagesimal 18 places of RSA-129:
42.46.29.41.37.48.55.25.30.05.42.37.50.46.52.42.05.41
18.17.16.15.14.13.12.11.10.09.08.07.06.05.04.03.02.01 - sexagesimal places
At p18 = 37.06.31.04.48.55.18.34.18.46.20.24.40.11.35.46.48.53
and q18 = 02.10.39.57.08.14.49.12.44.19.28.01.01.05.57.30.09.37
we find p18 x q18 = 1.20.48.49.54.55.41.18.45.42.22.41.05.10.25.05.04.55.42.46.29.41.37.48.55.25.30.05.42.37.50.46.52.42.05.41 .

16. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
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Step 9:
Make a dedicated multiplication table for
p18 = 37.06.31.04.48.55.18.34.18.46.20.24.40.11.35.46.48.53
q18 = 02.10.39.57.08.14.49.12.44.19.28.01.01.05.57.30.09.37
p18 x q18 = 1.20.48.49.54.55.41.18.45.42.22.41.05.10.25.05.04.55.42.46.29.41.37.48.55.25.30.05.42.37.50.46.52.42.05.41
Sieve the table for a match on the last 21 places of RSA-129:
05.58.23.42.46.29.41.37.48.55.25.30.05.42.37.50.46.52.42.05.41
21.20.19.18.17.16.15.14.13.12.11.10.09.08.07.06.05.04.03.02.01 - sexagesimal places
At p21 = 28.21.20.37.06.31.04.48.55.18.34.18.46.20.24.40.11.35.46.48.53
and q21 = 27.54.16.02.10.39.57.08.14.49.12.44.19.28.01.01.05.57.30.09.37
we find p21 x q21 = 13.11.15.03.58.34.52.25.36.52.59.56.31.03.19.18.19.09.37.43.18.05.58.23.42.46.29.41.37.48.55.25.30.05.42.37.50.46.52.42.05.41 .

17. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
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Step 10 :
Make a dedicated multiplication table for
p21 = 28.21.20.37.06.31.04.48.55.18.34.18.46.20.24.40.11.35.46.48.53
q21 = 27.54.16.02.10.39.57.08.14.49.12.44.19.28.01.01.05.57.30.09.37
p21 x q21 = 13.11.15.03.58.34.52.25.36.52.59.56.31.03.19.18.19.09.37.43.18.05.58.23.42.46.29.41.37.48.55.25.30.05.42.37.50.46.52.42.05.41 .
Sieve the table for a match on the last 24 sexagesimal places of RSA-129:
52.33.54.05.58.23.42.46.29.41.37.48.55.25.30.05.42.37.50.46.52.42.05.41
24.23.22.21.20.19.18.17.16.15.14.13.12.11.10.09.08.07.06.05.04.03.02.01 - sexagesimal places
At p24 = 01.29.23.28.21.20.37.06.31.04.48.55.18.34.18.46.20.24.40.11.35.46.48.53
and q24 = 26.24.05.27.54.16.02.10.39.57.08.14.49.12.44.19.28.01.01.05.57.30.09.37
we find p24 x q24 =
39.20.03.49.07.14.53.00.53.23.21.56.49.12.03.04.45.15.17.30.03.27.49.52.33.54.05.58.23.42.46.29.41.37.48.55.25.30.5.42.37.50.46.52.42.05.41 .

18. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
18
Step 11 :
Make a dedicated multiplication table for
p24 = 01.29.23.28.21.20.37.06.31.04.48.55.18.34.18.46.20.24.40.11.35.46.48.53
and q24 = 26.24.05.27.54.16.02.10.39.57.08.14.49.12.44.19.28.01.01.05.57.30.09.37
p24 x q24 = 39.20.03.49.07.14.53.00.53.23.21.56.49.12.03.04.45.15.17.30.03.27.49.52.33.54.05.58.23.42.46.29.41.37.48.55.25.30.5.42.37.50.46.52.42.05.41 .
Sieve the table for a match on the last 27 sexagesimal places of RSA-129:
59.46.14.52.33.54.05.58.23.42.46.29.41.37.48.55.25.30.05.42.37.50.46.52.42.05.41
27.26.25.24.23.22.21.20.19.18.17.16.15.14.13.12.11.10.09.08.07.06.05.04.03.02.01 - sexagesimal places
At p27 = 19.04.09.01.29.23.28.21.20.37.06.31.04.48.55.18.34.18.46.20.24.40.11.35.46.48.53
and q27 = 29.21.44.26.24.05.27.54.16.02.10.39.57.08.14.49.12.44.19.28.01.01.05.57.30.09.37
we find p27 x q27=
9.19.54.56.18.48.01.56.39.10.31.18.27.13.33.05.24.32.54.58.52.12.43.37.51.13.25.59.46.14.52.33.54.05.58.23.42.46.29.41.37.48.55.25.30.05.42.37.
50.46.52.42.05.41

19. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
19
Step 12 :
Make a dedicated multiplication table for
p27 = 19.04.09.01.29.23.28.21.20.37.06.31.04.48.55.18.34.18.46.20.24.40.11.35.46.48.53
q27 = 29.21.44.26.24.05.27.54.16.02.10.39.57.08.14.49.12.44.19.28.01.01.05.57.30.09.37
p27 x q27=
9.19.54.56.18.48.01.56.39.10.31.18.27.13.33.05.24.32.54.58.52.12.43.37.51.13.25.59.46.14.52.33.54.05.58.23.42.46.29.41.37.48.55.25.30.05.42.37.50.46.
52.42.05.41
Sieve the table for a match on the last 30 sexagesimal places of RSA-129:
49.09.09.59.46.14.52.33.54.05.58.23.42.46.29.41.37.48.55.25.30.05.42.37.50.46.52.42.05.41
30.29.28.27.26.25.24.23.22.21.20.19.18.17.16.15.14.13.12.11.10.09.08.07.06.05.04.03.02.01
At p30 = 16.42.49.19.04.09.01.29.23.28.21.20.37.06.31.04.48.55.18.34.18.46.20.24.40.11.35.46.48.53
and q30 = 15.01.47.29.21.44.26.24.05.27.54.16.02.10.39.57.08.14.49.12.44.19.28.01.01.05.57.30.09.37
we find p30 x q30 =
4.11.12.16.18.44.28.00.33.20.45.27.48.24.31.01.15.47.15.20.42.49.48.25.30.18.57.14.16.17.49.09.09.59.46.14.52.33.54.05.58.23.42.46.29.41.37.48.55.25.
30.05.42.37.50.46.52.42.05.41

20. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
20
Step 13 :
Make a dedicated multiplication table for
P30 = 16.42.49.19.04.09.01.29.23.28.21.20.37.06.31.04.48.55.18.34.18.46.20.24.40.11.35.46.48.53
q30 = 15.01.47.29.21.44.26.24.05.27.54.16.02.10.39.57.08.14.49.12.44.19.28.01.01.05.57.30.09.37
P30 x q30=
4.11.12.16.18.44.28.00.33.20.45.27.48.24.31.01.15.47.15.20.42.49.48.25.30.18.57.14.16.17.49.09.09.59.46.14.52.33.54.05.58.23.42.46.29.41.37.48.55.25.
30.05.42.37.50.46.52.42.05.41
Sieve the table for a match on the last 33 sexagesimal places of RSA-129:
46.02.23.49.09.09.59.46.14.52.33.54.05.58.23.42.46.29.41.37.48.55.25.30.05.42.37.50.46.52.42.05.41
33.32.31.30.29.28.27.26.25.24.23.22.21.20.19.18.17.16.15.14.13.12.11.10.09.08.07.06.05.04.03.02.01
At p33 = 18.05.39.16.42.49.19.04.09.01.29.23.28.21.20.37.06.31.04.48.55.18.34.18.46.20.24.40.11.35.46.48.53
and q33 = 5.01.51.15.01.47.29.21.44.26.24.05.27.54.16.02.10.39.57.08.14.49.12.44.19.28.01.01.05.57.30.09.37
we find p33 x q33 =
1.31.01.49.23.11.15.55.37.31.52.11.59.13.54.59.35.57.20.16.17.48.40.31.31.29.58.54.32.34.18.22.59.46.02.23.49.09.09.59.46.14.52.33.54.05.58.23.42.46.0
29.41.37.48.55.25.30.05.42.37.50.46.52.42.05.41

21. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
21
Step 14 :
Make a dedicated multiplication table for
At p33 = 18.05.39.16.42.49.19.04.09.01.29.23.28.21.20.37.06.31.04.48.55.18.34.18.46.20.24.40.11.35.46.48.53
and q33 = 5.01.51.15.01.47.29.21.44.26.24.05.27.54.16.02.10.39.57.08.14.49.12.44.19.28.01.01.05.57.30.09.37
we find p33 x q33=
1.31.01.49.23.11.15.55.37.31.52.11.59.13.54.59.35.57.20.16.17.48.40.31.31.29.58.54.32.34.18.22.59.
46.02.23.49.09.09.59.46.14.52.33.54.05.58.23.42.46.029.41.37.48.55.25.30.05.42.37.50.46.52.42.05.41
Sieve the table for a match on the last 36 sexagesimal places of RSA-129:
02.30.52.46.02.23.49.09.09.59.46.14.52.33.54.05.58.23.42.46.29.41.37.48.55.25.30.05.42.37.50.46.52.42.05.41
36.35.34.33.32.31.30.29.28.27.26.25.24.23.22.21.20.19.18.17.16.15.14.13.12.11.10.09.08.07.06.05.04.03.02.01
At p36 = 10.37.16.18.05.39.16.42.49.19.04.09.01.29.23.28.21.20.37.06.31.04.48.55.18.34.18.46.20.24.40.11.35.46.48.53
and q36 = 20.18.17.05.01.51.15.01.47.29.21.44.26.24.05.27.54.16.02.10.39.57.08.14.49.12.44.19.28.01.01.05.57.30.09.37
we find p36 x q36 =
3.35.39.38.22.21.58.04.40.52.19.39.33.54.34.24.8.10.52.33.27.38.22.46.36.27.23.59.27.02.38.21.50.15.14.35.02.30.52.46.02.23.49.09.09.59.46.14.52.33.54.
05.58.23.42.46.29.41.37.48.55.25.30.05.42.37.50.46.52.42.05.41

22. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
22
Step 15 :
Make a dedicated multiplication table for
p36 = 10.37.16.18.05.39.16.42.49.19.04.09.01.29.23.28.21.20.37.06.31.04.48.55.18.34.18.46.20.24.40.11.35.46.48.53
q36 = 20.18.17.05.01.51.15.01.47.29.21.44.26.24.05.27.54.16.02.10.39.57.08.14.49.12.44.19.28.01.01.05.57.30.09.37
p36 x q36 =
3.35.39.38.22.21.58.04.40.52.19.39.33.54.34.24.08.10.52.33.27.38.22.46.36.27.23.59.27.02.38.21.50.15.14.35.02.30.52.46.02.23.49.09.09.59.46.14.52.33.0
54.05.58.23.42.46.29.41.37.48.55.25.30.05.42.37.50.46.52.42.05.41
Extend p and q by 601
.
Sieve the table on the last 37 sexagesimal places of RSA-129b60:
26.02.30.52.46.02.23.49.09.09.59.46.14.52.33.54.05.58.23.42.46.29.41.37.48.55.25.30.05.42.37.50.46.52.42.05.41
37.36.35.34.33.32.31.30.29.28.27.26.25.24.23.22.21.20.19.18.17.16.15.14.13.12.11.10.09.08.07.06.05.04.03.02.01

23. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
23
A match -in the first row (q36)- at
p37 = 03.10.37.16.18.05.39.16.42.49.19.04.09.01.29.23.28.21.20.37.06.31.04.48.55.18.34.18.46.20.24.40.11.35.46.48.53 .
q37 = 00.20.18.17.05.01.51.15.01.47.29.21.44.26.24.05.27.54.16.02.10.39.57.08.14.49.12.44.19.28.01.01.05.57.30.09.37 .
p37 x q37 =
1.04.30.30.53.27.55.43.10.03.20.24.52.53.06.50.47.50.58.59.05.27.29.47.31.04.05.36.57.51.05.41.39.42.45.43.26.02.30.52.46.02.23.49.09.09.59.46.14.52.3
3.54.05.58.23.42.46.29.41.37.48.55.25.30.05.42.37.50.46.52.42.05.41 = RSA-129 base 60
Step 16
P37 x q36 = RSA-129 base 60
This is the procedure.

24. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
24
Annex A
The Babylonian way of factoring RSA-129
Factorization of RSA-129 base 60, number of steps of the procedure.
Step Description Product Formula steps Total
1 RSA-129 to RSA-129 base 60 RSA-129 base 60
2 Note the .41 entries from the
Std. Prime Multiplication Table
8 entries,
p x q =.41
3 Construct 8 tables p x q = .41, p
and q 6 places
8 x (605
)2
8 x 6010
4 Sieve the tables for the last 6
places of RSA
P6 = .53 and
q6 = .37 8 x (605
)2
8 x 6010
5 Construct dedicated table Table p6 x q6 1 x (603
)2
606
Sieve the table for the last 9
places of RSA p9 x q9 = RSA9 1 x (603
606
6 Construct dedicated table Table p9 x q9 1 x (603
)2
606
Sieve the table for the last 12
places of RSA p12 x q12 = RSA12 1 x (603
606
7 Construct dedicated table Table p12 x q12 1 x (603
)2
606
Sieve the table for the last 15
places of RSA p15 x q15 = RSA15 1 x (603
606
8 Construct dedicated table Table p15 x q15 1 x (603
)2
606
Sieve the table for the last 18
places of RSA p18 x q18 = RSA18 1 x (603
606
9 Construct dedicated table p18 x q18 = RSA18 1 x (603
)2
606
Sieve the table for the last 21
places of RSA p21 x q21 = RSA21 1 x (603
)2
8 x 606
10 Construct dedicated table Table p21 x q21 1 x (603
)2
606
Sieve the table for the last 24
places of RSA p24 x q24 = RSA24 1 x (603
606

25. The Babylonian way of factoring RSA - The Foundation of Mathematics - M.W.R. Karskens 2019
25
11 Construct dedicated table Table p24 x q24 1 x (603
)2
606
Sieve the table for the last 27
places of RSA p27 x q27 = RSA27 1 x (603
606
12 Construct dedicated table Table p27 x q27 1 x (603
)2
606
Sieve the table for the last 30
places of RSA p30 x q30 = RSA30 1 x (60)3
606
13 Construct dedicated table Table p30 x q30 1 x (603
)2
606
Sieve the table for the last 33
places of RSA p33 x q33 = RSA33 1 x (603
606
14 Construct dedicated table Table p33 x q33 1 x (603
)2
606
Sieve the table for the last 36
places of RSA p36 x q36 = RSA36 1 x (603
606
15 Construct dedicated table Table p37 x q37 1 x (601)2
602
Sieve the table for the last 37
places of RSA p37 x q37 = RSA37 3 x 1 3
16 p37 x q36 = RSA-129b60
(16 x 6010
) + 22 x 606
) +
602
Total steps =
(16 x 6010
) + (22 x 606
) + 602
= 16.00.00.00.00.22.00.00.00.01.00.00 base 60
= 580475290626432003600 base 10
Some final remarks
• If construction and sieving count as one step, multiply by 8 instead of 16 and 11 instead of 22.
• For tables .01 and .49 -multiply by 12 instead of 8.
• For RSA numbers with as their last sexagesimal place .01 and .49, you multiply with 24 instead of 16 .
• For RSA numbers larger than RSA-129, the factor ‘22’ increases; for RSA-2048 it is ‘55’ .
• The choice of three sexagesimal places for the dedicated multiplication is arbitrary.