View Full Version : Find (A,B,W,X,Y,Z) such that ...

Richard Amiel McGough

11-18-2011, 01:46 PM

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

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, 07:05 PM

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:

<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 />" + A.toString() + B.toString() + " x " + B.toString() + A.toString() + " = " + prod.toString();

sum = prod + Number(reverseString(prod.toString()));

lst += "" + sum + "";

}

}

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, 08:16 PM

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:

<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 />" + A.toString() + B.toString() + " x " + B.toString() + A.toString() + " = " + prod.toString();

sum = prod + Number(reverseString(prod.toString()));

lst += "" + sum + "";

}

}

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. :signthankspin:

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, 08:56 PM

Thanks for doing the calculations, Richard. :signthankspin:

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, 09:24 PM

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, 11:24 AM

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:

<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 />" + A.toString() + B.toString() + " x " + B.toString() + A.toString() + " = " + prod.toString();

sum = prod + Number(reverseString(prod.toString()));

lst += "" + sum + "";

}

}

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, 01:20 PM

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, 03:11 PM

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. :thumb:

Richard Amiel McGough

11-20-2011, 08:50 PM

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. :thumb:

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, 09:49 PM

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...

Richard Amiel McGough

11-20-2011, 09:56 PM

"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...

You are correct. "Palindromic" isn't the right word because it refers to a single string that reads the same forwards and backwards like abcba. The pair 37/73 are emirps or "reversible primes." The fact that they are primes is an "added bonus" - it is conceivable that they could have been some pair of composite numbers, and primes are special. Indeed, they are prime (as in premier) amongst the integers.

RC Christian

11-21-2011, 12:02 AM

You are correct. "Palindromic" isn't the right word because it refers to a single string that reads the same forwards and backwards like abcba. The pair 37/73 are emirps or "reversible primes." The fact that they are primes is an "added bonus" - it is conceivable that they could have been some pair of composite numbers, and primes are special. Indeed, they are prime (as in premier) amongst the integers.

Premier primes...cool man :thumb:

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