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

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

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.
Last edited by RAM; 11-18-2011 at 01:20 PM. Reason: RC Christian pointed out an error. Thanks!

2. 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>

3. Thanks

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.

4. 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.

5. 2

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.

6. ABC x CBA = UVWXYZ

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

7. 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.

8. Thank you

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.

9. 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

10. 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...

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