Find (A,B,W,X,Y,Z) such that ...

**Richard Amiel McGough** · 11-18-2011, 12:46 PM

Our friend RC Christian posed an interesting mathematical challenge to a professor:

Originally Posted by RC Christian

On of the professors from the list, a Wofford College Ph.D mathematician found one little jewel that he took the time to look at, and it was one of the points that I really didn't find all that interesting:

37 x 73 = 2701 --> 2701 + 1072 (reflective anagram) = 3773, the combined prime factors. My challenge on this was for him to attempt to find another number that had this distinctive characteristic. He was "intrigued", and stated that he tried, but couldn't. But beyond that, he said, "this is basically numerology and I'm not interested, but if it somehow helps to strength you're faith, I see merit in it there." Strength my faith???? Numerology??? The patterns are either there or not their not, and if they are there, wouldn't this be a kind of important finding for the world to know about? I'm mean, if hardly anyone had ever heard of this book, called the Holy Bible, yeah, so what, but...

This challenge can be formalized as follows:

Find six digits (A,B,W,X,Y,Z) in any base such that AB x BA= WXYZ and WXYZ + ZYXW = ABBA

Note: AB and WXYZ represent standard numerical notation in base b, e.g. WXYZ = Wb³ + Xb² + Yb¹ + Zb⁰.

**Richard Amiel McGough** · 11-18-2011, 06:05 PM

Originally Posted by RAM

Our friend RC Christian posed an interesting mathematical challenge to a professor:

This challenge can be formalized as follows:

Find six digits (A,B,W,X,Y,Z) in any base such that AB x BA= WXYZ and WXYZ + ZYXW = ABBA

Note: AB and WXYZ represent standard numerical notation in base b, e.g. WXYZ = Wb³ + Xb² + Yb¹ + Zb⁰.

Well, working out the abstract analysis is a little complex, so I just wrote a quick script to test this in base 10. Here are the results. It is interesting that this algorythm produces the same number for the sum of the product and its reverse. The number 3773 appears twice, and 10890 appears six times. The table is exhaustive, so there are is no other two digit set like (37, 73) that satisfies the conditions.

11 x 11 = 121	242
12 x 21 = 252	504
13 x 31 = 403	707
14 x 41 = 574	1049
15 x 51 = 765	1332
16 x 61 = 976	1655
17 x 71 = 1207	8228
18 x 81 = 1458	9999
19 x 91 = 1729	11000
22 x 22 = 484	968
23 x 32 = 736	1373
24 x 42 = 1008	9009
25 x 52 = 1300	1331
26 x 62 = 1612	3773
27 x 72 = 1944	6435
28 x 82 = 2296	9218
29 x 92 = 2668	11330
33 x 33 = 1089	10890
34 x 43 = 1462	4103
35 x 53 = 1855	7436
36 x 63 = 2268	10890
37 x 73 = 2701	3773
38 x 83 = 3154	7667
39 x 93 = 3627	10890
44 x 44 = 1936	8327
45 x 54 = 2430	2772
46 x 64 = 2944	7436
47 x 74 = 3478	12221
48 x 84 = 4032	6336
49 x 94 = 4606	10670
55 x 55 = 3025	8228
56 x 65 = 3640	4103
57 x 75 = 4275	9999
58 x 85 = 4930	5324
59 x 95 = 5605	10670
66 x 66 = 4356	10890
67 x 76 = 5092	7997
68 x 86 = 5848	14333
69 x 96 = 6624	10890
77 x 77 = 5929	15224
78 x 87 = 6786	13662
79 x 97 = 7663	11330
88 x 88 = 7744	12221
89 x 98 = 8722	11000
99 x 99 = 9801	10890

And here is the html javascript page I wrote to do this calculation:

Code:

<html>
<script type="text/javascript">
function calc(){
   var lst = "";
   b = 10; 
   for (var A=1;A<b;A++) {
    for (var B=A;B<b;B++) {
 var prod = (A * b + B)*(B*b + A);
        lst += "<br />[tr][td]" + A.toString() + B.toString() + " x " + B.toString() + A.toString() + " = " + prod.toString();
 sum = prod + Number(reverseString(prod.toString()));
 lst += "[/td][td]" + sum + "[/td][/tr]";  
    }
} 
document.getElementById('lulu').innerHTML="";
}
function reverseString(str) { 
 var a=str.split(''), b = a.length; 
 for (var i=0; i<b; i++) { 
      a.unshift(a.splice(1+i,1).shift()) 
 } 
 a.shift();
 return a.join(''); 
} 
</script>
<body>
<Input id="calc" name="calc" type="submit" onclick="calc()" value="Do calc">
 
<div id="lulu">
This is where lst will go
</div> 
</body>
</html>

**RC Christian** · 11-18-2011, 07:16 PM

Originally Posted by RAM

Well, working out the abstract analysis is a little complex, so I just wrote a quick script to test this in base 10. Here are the results. It is interesting that this algorythm produces the same number for the sum of the product and its reverse. The number 3773 appears twice, and 10890 appears six times. The table is exhaustive, so there are is no other two digit set like (37, 73) that satisfies the conditions.

11 x 11 = 121	242
12 x 21 = 252	504
13 x 31 = 403	707
14 x 41 = 574	1049
15 x 51 = 765	1332
16 x 61 = 976	1655
17 x 71 = 1207	8228
18 x 81 = 1458	9999
19 x 91 = 1729	11000
22 x 22 = 484	968
23 x 32 = 736	1373
24 x 42 = 1008	9009
25 x 52 = 1300	1331
26 x 62 = 1612	3773
27 x 72 = 1944	6435
28 x 82 = 2296	9218
29 x 92 = 2668	11330
33 x 33 = 1089	10890
34 x 43 = 1462	4103
35 x 53 = 1855	7436
36 x 63 = 2268	10890
37 x 73 = 2701	3773
38 x 83 = 3154	7667
39 x 93 = 3627	10890
44 x 44 = 1936	8327
45 x 54 = 2430	2772
46 x 64 = 2944	7436
47 x 74 = 3478	12221
48 x 84 = 4032	6336
49 x 94 = 4606	10670
55 x 55 = 3025	8228
56 x 65 = 3640	4103
57 x 75 = 4275	9999
58 x 85 = 4930	5324
59 x 95 = 5605	10670
66 x 66 = 4356	10890
67 x 76 = 5092	7997
68 x 86 = 5848	14333
69 x 96 = 6624	10890
77 x 77 = 5929	15224
78 x 87 = 6786	13662
79 x 97 = 7663	11330
88 x 88 = 7744	12221
89 x 98 = 8722	11000
99 x 99 = 9801	10890

And here is the html javascript page I wrote to do this calculation:

Code:

<html>
<script type="text/javascript">
function calc(){
   var lst = "";
   b = 10; 
   for (var A=1;A<b;A++) {
    for (var B=A;B<b;B++) {
 var prod = (A * b + B)*(B*b + A);
        lst += "<br />[tr][td]" + A.toString() + B.toString() + " x " + B.toString() + A.toString() + " = " + prod.toString();
 sum = prod + Number(reverseString(prod.toString()));
 lst += "[/td][td]" + sum + "[/td][/tr]";  
    }
} 
document.getElementById('lulu').innerHTML="";
}
function reverseString(str) { 
 var a=str.split(''), b = a.length; 
 for (var i=0; i<b; i++) { 
      a.unshift(a.splice(1+i,1).shift()) 
 } 
 a.shift();
 return a.join(''); 
} 
</script>
<body>
<Input id="calc" name="calc" type="submit" onclick="calc()" value="Do calc">
 
<div id="lulu">
This is where lst will go
</div> 
</body>
</html>

Thanks for doing the calculations, Richard.

On a separate note, I understand there's two schools of thought on whether the Babylonians, Egyptians, Hebrews, etc., considered 1 or 2 to be the first prime number. I've read a couple of older posts where there were some remarks on which number was the first prime. What's you're take on it.

**Richard Amiel McGough** · 11-18-2011, 07:56 PM

Originally Posted by RC Christian

Thanks for doing the calculations, Richard.

On a separate note, I understand there's two schools of thought on whether the Babylonians, Egyptians, Hebrews, etc., considered 1 or 2 to be the first prime number. I've read a couple of older posts where there were some remarks on which number was the first prime. What's you're take on it.

That's a great question. I know Euclid proved there aer infinitely many primes around 300 BC, so we know the ancients knew about them. But I don't know when the whole question about the number 1 came up. But there are mathematical reasons to distinguish between primes and the Unit (1) so it sounds like a question that probably wasn't even thought of till maybe the Renaissance or thereabouts. It sounds like too sophisticated of a question for the "ancients." But now that the question is in my head an answer is sure to come.

I made a big spreadsheet of the first few hundred primes and looked for correlations between index and prime depending on whether I started with 1 or 2. I never noticed any sufficiently strong correlations to convince me either was supperior. But there are a few things I like about starting with 1. I like the self-reflective coherence of the first three index/prime pairs and the fact that the 7th pair is a hex/star pair. And we have two star numbers in the 13th pair.

1/1
2/2/
3/3
4/5
5/7
6/11
7/13 = Hex(2)/Star(2)
8/17
9/19
10/23
11/29
12/31
13/37 = Star(2)/Star(3)
14/41

But it doesn't make any sense to me to haggle over the "definition" of the number 1 as a prime. It would affect no mathematical results, so why worry about it? You don't need to justify it if you want to start indexing primes from the number 1. Just do it. You are free and no one can say that you are wrong because it would not lead to any logical contradictions. Therefore it is mathematically permissible.

**RC Christian** · 11-18-2011, 08:24 PM

Originally Posted by RAM

That's a great question. I know Euclid proved there aer infinitely many primes around 300 BC, so we know the ancients knew about them. But I don't know when the whole question about the number 1 came up. But there are mathematical reasons to distinguish between primes and the Unit (1) so it sounds like a question that probably wasn't even thought of till maybe the Renaissance or thereabouts. It sounds like too sophisticated of a question for the "ancients." But now that the question is in my head an answer is sure to come.

I made a big spreadsheet of the first few hundred primes and looked for correlations between index and prime depending on whether I started with 1 or 2. I never noticed any sufficiently strong correlations to convince me either was supperior. But there are a few things I like about starting with 1. I like the self-reflective coherence of the first three index/prime pairs and the fact that the 7th pair is a hex/star pair. And we have two star numbers in the 13th pair.

1/1
2/2/
3/3
4/5
5/7
6/11
7/13 = Hex(2)/Star(2)
8/17
9/19
10/23
11/29
12/31
13/37 = Star(2)/Star(3)
14/41

But it doesn't make any sense to me to haggle over the "definition" of the number 1 as a prime. It would affect no mathematical results, so why worry about it? You don't need to justify it if you want to start indexing primes from the number 1. Just do it. You are free and no one can say that you are wrong because it would not lead to any logical contradictions. Therefore it is mathematically permissible.

Cool. Thanks for the reply and your the explanation of why you like starting with 1. I'm a "2" man, personally. Over the weekend I'll try to post some of the reasons I like starting with 2, when indexing primes.

**RC Christian** · 11-20-2011, 10:24 AM

Originally Posted by RAM

Well, working out the abstract analysis is a little complex, so I just wrote a quick script to test this in base 10. Here are the results. It is interesting that this algorythm produces the same number for the sum of the product and its reverse. The number 3773 appears twice, and 10890 appears six times. The table is exhaustive, so there are is no other two digit set like (37, 73) that satisfies the conditions.

11 x 11 = 121	242
12 x 21 = 252	504
13 x 31 = 403	707
14 x 41 = 574	1049
15 x 51 = 765	1332
16 x 61 = 976	1655
17 x 71 = 1207	8228
18 x 81 = 1458	9999
19 x 91 = 1729	11000
22 x 22 = 484	968
23 x 32 = 736	1373
24 x 42 = 1008	9009
25 x 52 = 1300	1331
26 x 62 = 1612	3773
27 x 72 = 1944	6435
28 x 82 = 2296	9218
29 x 92 = 2668	11330
33 x 33 = 1089	10890
34 x 43 = 1462	4103
35 x 53 = 1855	7436
36 x 63 = 2268	10890
37 x 73 = 2701	3773
38 x 83 = 3154	7667
39 x 93 = 3627	10890
44 x 44 = 1936	8327
45 x 54 = 2430	2772
46 x 64 = 2944	7436
47 x 74 = 3478	12221
48 x 84 = 4032	6336
49 x 94 = 4606	10670
55 x 55 = 3025	8228
56 x 65 = 3640	4103
57 x 75 = 4275	9999
58 x 85 = 4930	5324
59 x 95 = 5605	10670
66 x 66 = 4356	10890
67 x 76 = 5092	7997
68 x 86 = 5848	14333
69 x 96 = 6624	10890
77 x 77 = 5929	15224
78 x 87 = 6786	13662
79 x 97 = 7663	11330
88 x 88 = 7744	12221
89 x 98 = 8722	11000
99 x 99 = 9801	10890

And here is the html javascript page I wrote to do this calculation:

Code:

<html>
<script type="text/javascript">
function calc(){
   var lst = "";
   b = 10; 
   for (var A=1;A<b;A++) {
    for (var B=A;B<b;B++) {
 var prod = (A * b + B)*(B*b + A);
        lst += "<br />[tr][td]" + A.toString() + B.toString() + " x " + B.toString() + A.toString() + " = " + prod.toString();
 sum = prod + Number(reverseString(prod.toString()));
 lst += "[/td][td]" + sum + "[/td][/tr]";  
    }
} 
document.getElementById('lulu').innerHTML="";
}
function reverseString(str) { 
 var a=str.split(''), b = a.length; 
 for (var i=0; i<b; i++) { 
      a.unshift(a.splice(1+i,1).shift()) 
 } 
 a.shift();
 return a.join(''); 
} 
</script>
<body>
<Input id="calc" name="calc" type="submit" onclick="calc()" value="Do calc">
 
<div id="lulu">
This is where lst will go
</div> 
</body>
</html>

Richard,

Would you mind tweaking javascript to solve for:

Find 9 digits (ABCUVWXYZ), such that, ABC X CBA = UVWXYZ and UVWXYZ + ZYXWVU = ABCCBA

I'm pretty sure there aren't, but it would be interesting to know (and is the challenge the Wofford professor tossed back to me).

Thanks

**Richard Amiel McGough** · 11-20-2011, 12:20 PM

Originally Posted by RC Christian

Richard,

Would you mind tweaking javascript to solve for:

Find 9 digits (ABCUVWXYZ), such that, ABC X CBA = UVWXYZ and UVWXYZ + ZYXWVU = ABCCBA

I'm pretty sure there aren't, but it would be interesting to know (and is the challenge the Wofford professor tossed back to me).

Thanks

I did the calculation, and did not find any three digit combinations that satisfied the conditions. The closest I found was this:

990 x 099 = 98010 and 98010 + 01089 = 99099

The properties of 37/73 are both beautiful and very rare. Especially when seen in the context of the Holographic Generating Set (A = 27, B = 37, C = 73, D = 137) where D = A + B + C. This set is profoundly integrated with (actually derived from) the unified alphanumeric structure of Genesis 1:1-5 & John 1:1-5, which I call the Full Creation Hyper-Holograph. Here are a few identities. The first verse has an elaborate substructure that can be expressed in four different was as combinations of the elements of the GenSet (where Tri(n) = nth triangular number):

Gen 1:1 = 2701	= Tri(C)
	= BC
	= 100A + 1
	= Tri(B) + 3Tri(B-1)

Remember, the final identity is derived from the natural syntax of the versse because "and the earth" = 703 = Tri(37) = Tri(B).

And both Gen 1:1 and John 1:1 are the products of symmetrical factors and can be expressed using different elements of the GenSet:

Gen 1:1	= 2701 = 37 x 73	= 100A + 1
John 1:1	= 3627 = 39 x 93	= 100B - C

This means that Gen 1:1 and John 1:1 are linked via the elements of the GenSet:

Gen 1:1 + AB = John 1:1 + C

And the common value? It is 3700 = 100B = 27 x 137 + 1 so too can be expressed in two different ways using elements from the GenSet:

Gen 1:1 + AB = 100B = AD + 1 = John 1:1 + C

I could go on. There seems to be no limit to the endless fractal harmony of the GenSet derived from those two premier "creation passages" in the Bible.

**RC Christian** · 11-20-2011, 02:11 PM

Originally Posted by RAM

I did the calculation, and did not find any three digit combinations that satisfied the conditions. The closest I found was this:

990 x 099 = 98010 and 98010 + 01089 = 99099

The properties of 37/73 are both beautiful and very rare. Especially when seen in the context of the Holographic Generating Set (A = 27, B = 37, C = 73, D = 137) where D = A + B + C. This set is profoundly integrated with (actually derived from) the unified alphanumeric structure of Genesis 1:1-5 & John 1:1-5, which I call the Full Creation Hyper-Holograph. Here are a few identities. The first verse has an elaborate substructure that can be expressed in four different was as combinations of the elements of the GenSet (where Tri(n) = nth triangular number):

Gen 1:1 = 2701	= Tri(C)
	= BC
	= 100A + 1
	= Tri(B) + 3Tri(B-1)

Remember, the final identity is derived from the natural syntax of the versse because "and the earth" = 703 = Tri(37) = Tri(B).

And both Gen 1:1 and John 1:1 are the products of symmetrical factors and can be expressed using different elements of the GenSet:

Gen 1:1	= 2701 = 37 x 73	= 100A + 1
John 1:1	= 3627 = 39 x 93	= 100B - C

This means that Gen 1:1 and John 1:1 are linked via the elements of the GenSet:

Gen 1:1 + AB = John 1:1 + C

And the common value? It is 3700 = 100B = 27 x 137 + 1 so too can be expressed in two different ways using elements from the GenSet:

Gen 1:1 + AB = 100B = AD + 1 = John 1:1 + C

I could go on. There seems to be no limit to the endless fractal harmony of the GenSet derived from those two premier "creation passages" in the Bible.

Thanks, Richard, for taking the time to carry out the calculations. Do you think it would be rather safe to say that 37 / 73 possibly are the only 2 reflective integer anagrams of any order that behave in this fashion? And if so, how beautiful it is that those numbers are the ordinal and standard values of "chokmah", respectively. I really admire the holographic work you've done on Gen 1:1-5 and John 1:1-5, and have spent a lot of time a year or so ago studying it.

Referring back to the initial javascript you ran on AB X BA = WXYZ, I (personally) find it significant that 26 x 62 = 3773 (the only other factors that equal 3773 in this arrangement) and then John 1:1-5 equals, in gematria, 23088 (26 X 888), with John 1:1 sharing a similar type of reflective anagram identity (39 X 93) with Genesis 1:1.

I have some other unique identities about both of these passages that I'll try to post tonight...they tie into the prime # posting that I've been trying to get to. Thanks for your time and excellent work.

**Richard Amiel McGough** · 11-20-2011, 07:50 PM

Originally Posted by RC Christian

Thanks, Richard, for taking the time to carry out the calculations. Do you think it would be rather safe to say that 37 / 73 possibly are the only 2 reflective integer anagrams of any order that behave in this fashion? And if so, how beautiful it is that those numbers are the ordinal and standard values of "chokmah", respectively. I really admire the holographic work you've done on Gen 1:1-5 and John 1:1-5, and have spent a lot of time a year or so ago studying it.

Referring back to the initial javascript you ran on AB X BA = WXYZ, I (personally) find it significant that 26 x 62 = 3773 (the only other factors that equal 3773 in this arrangement) and then John 1:1-5 equals, in gematria, 23088 (26 X 888), with John 1:1 sharing a similar type of reflective anagram identity (39 X 93) with Genesis 1:1.

I have some other unique identities about both of these passages that I'll try to post tonight...they tie into the prime # posting that I've been trying to get to. Thanks for your time and excellent work.

Hey there RC,

Yes, I think it's more than "safe to say" that 37 / 73 is the only palindromic pair of two digit primes that have the property we've been discussing (in base 10, anyway). And yes, their relation to the ordinal and standard values of the Hebrew Hokmah (Wisdom) is quite beautiful, and this is augmented by the verses that speak of God creating by "wisdom" and that connect the word "wisdom" with "reshit" beginning which led ancient rabbinic interpretors to say that bereshit (in the begininng) means "with wisdom." There's really know end to the beauty or depth of these relations.

I look forward to reviewing the other identities you have to share.

Richard

**RC Christian** · 11-20-2011, 08:49 PM

Originally Posted by RAM

Hey there RC,

Yes, I think it's more than "safe to say" that 37 / 73 is the only palindromic pair of two digit primes that have the property we've been discussing (in base 10, anyway). And yes, their relation to the ordinal and standard values of the Hebrew Hokmah (Wisdom) is quite beautiful, and this is augmented by the verses that speak of God creating by "wisdom" and that connect the word "wisdom" with "reshit" beginning which led ancient rabbinic interpretors to say that bereshit (in the begininng) means "with wisdom." There's really know end to the beauty or depth of these relations.

I look forward to reviewing the other identities you have to share.

Richard

"37 / 73 is the only palindromic pair of two digit primes..." ? We can say a little more than that, can't we? Maybe, 37 / 73 is the only pair of two or three digit "reflective anagrams" (not exactly mathematical nomenclature) that have the property...