Elevens!

[Kees]

Obviously, I was applying the second rule

[mfgoode]

QUOTE=mfgoode]:surprised
Thats okay with me R.D. IT suits me fine. Good riddance to bad rubbish. I dont need you anyway as you seldom solve anything in my opinion. As Dr. Peter would say 'You have reached your level of incompetence' and you intend staying there for quite awhile.
The venom you tend to exude with every post will consume you.
No hard feelings and I wish you luck
Mally [/QUOTE


I'm sorry for those remarks R.D.
When the super heated steam engine exceeds the designed value (220 lbs / sq. in. in my Dad's days) a safety valve blows to release the excess and reduce the pressure so that the boiler does not explode.
Thats exactly what happened to me and I straight away let off steam.
I retract my vitriolic remarks and kindly accept my apologies.
I know it gets lonely at the top
Just to remind you that we all need and admire you for your above average knowledge which is exceptional and keep the fire burning. Even one candle can expel all the darkness that surrounds us all the time.
Mally

[mfgoode]

Originally Posted by Kees

Obviously, I was applying the second rule

Thank you Kees.
Well this thread wont be complete and exhausted without this last problem though I am sure there may be many more.
here are the nine digits so arranged that they form four square numbers :
9 , 81 , 324 , 576. (using all 9 digits and no more repetitions)
Now could you put them all together so as to form a single square ?
1) the smallest possible ? and
2) the largest possible ?
Mally

[alpertron]

With this simple program I wrote in UBASIC:

  10   dim C(9)
  20   for A=10000 to 31622
  30   B=A*A
  40   C(1)=B@10
  50   I=2
  60   C(I)=(B\10^(I-1))@10
  70   if C(I)=0 then 150
  80   J=1
  90   if C(I)=C(J) then 150
 100   J=J+1
 110   if J<I then 90
 120   I=I+1
 130   if I<10 then 60
 140   print A,B
 150   next A

I found the 30 squares formed with the digits 1-9:

 11826 	 139854276 
 12363 	 152843769 
 12543 	 157326849 
 14676 	 215384976 
 15681 	 245893761 
 15963 	 254817369 
 18072 	 326597184 
 19023 	 361874529 
 19377 	 375468129 
 19569 	 382945761 
 19629 	 385297641 
 20316 	 412739856 
 22887 	 523814769 
 23019 	 529874361 
 23178 	 537219684 
 23439 	 549386721 
 24237 	 587432169 
 24276 	 589324176 
 24441 	 597362481 
 24807 	 615387249 
 25059 	 627953481 
 25572 	 653927184 
 25941 	 672935481 
 26409 	 697435281 
 26733 	 714653289 
 27129 	 735982641 
 27273 	 743816529 
 29034 	 842973156 
 29106 	 847159236 
 30384 	 923187456

It appears that no combination of the numbers 9 , 81 , 324 , 576 are in these squares.

[mfgoode]

:surprised
Excellent Alpertron!
But what about the smallest square using all the 9 digits? I dont see it around in your post. Thanks!
Mally

[alpertron]

The smallest is 11826² = 139854276 and the greatest is 30384² = 923187456.

[mfgoode]

Originally Posted by alpertron

The smallest is 11826² = 139854276 and the greatest is 30384² = 923187456.

Thanks once again. How stupid of me! It is sitting right on top of your program. I realised this when I quit mersenneforum. By then it was too late to edit my post as the time limit was up.
You have made an interesting point in the same post #15. If you multiply 9,81,324,576 you will get a square naturally but the digits 2,5,7 are missing and so it does not meet the criterion..
Also I came across how to tell a probable square by inspection no matter how Large it is we must see if the last number is one of the foll:
1,4,9,6,5 and it goes on in the reverse sequence as 5,6,9,4,1 and so on and on. Of course this is elementary but few people use this.
In A run through of your program you have given, we can spot a mistake just by checking if the last number is one of those mentioned. Any other digit will highlight that it cannot be a perfect square.
Thanks Alpertron,
Mally
P.S. BTW How far do tables on squares go? a million? or 10 million of them? This will be easy by your programs to tabulate them. I mean somehing like Lehmer's table of primes which can be used as a ready reference for those who need it
As You are onto sequences, perhaps you could add these on as part of your
web site and repertoire. Only a suggestion.

[alpertron]

Originally Posted by mfgoode

Also I came across how to tell a probable square by inspection no matter how Large it is we must see if the last number is one of the foll:
1,4,9,6,5 and it goes on in the reverse sequence as 5,6,9,4,1 and so on and on. Of course this is elementary but few people use this.

The number 576 = 24² is a perfect square. According to your last paragraph, it ends in 6, so let's continue. But the second digit from the right is not on your list. So it appears that something is wrong.

Originally Posted by mfgoode

P.S. BTW How far do tables on squares go? a million? or 10 million of them? This will be easy by your programs to tabulate them. I mean somehing like Lehmer's table of primes which can be used as a ready reference for those who need it
As You are onto sequences, perhaps you could add these on as part of your
web site and repertoire. Only a suggestion.

Using a computer you can find if a number of 10000 digits is a perfect square in a fraction of second (using clever programming, of course). So it is pointless to store tables of squares, unless they also have other interesting properties.

[mfgoode]

Originally Posted by alpertron

The number 576 = 24² is a perfect square. According to your last paragraph, it ends in 6, so let's continue. But the second digit from the right is not on your list. So it appears that something is wrong.

Using a computer you can find if a number of 10000 digits is a perfect square in a fraction of second (using clever programming, of course). So it is pointless to store tables of squares, unless they also have other interesting properties.

Squares and factorisation.
The last digits 0, I , 4 , 5 , 6, 9 are okay for a cursory check, but not enough to be reasonably sure of the number being a perfect square.
With some factorisation methods and problems in number theory, it is of importance to decide quickly whether a number can be a perfect square.
I therefore give the last two digits in a square number.
These digits are limited to the following 22 possibilities.
00 21 41 64 89
01 24 44 69 96
04 25 49 76
09 29 56 81
16 36 61 84

Another useful observation is that when the smallest prime factor p of a number is found to be greater than the cube root [3/rt.] of n the other factor m in n=p*m must be prime.
Thus if m = a*b were composite both a and b would exceed the cube root n and one would obtain the contradiction
n=p*.a*.b > (3/rt).* n *(3/rt). n *(3/rt)*.n = n

Eg: Find the prime factorisation of n = 377,161.
Find sq.rt. n < 614.
The smallest factor is p=137 and n=137*2753.
But (3/rt). n < 73.
Hence 2753 is a prime.
This may be old hat to you but I’m putting it down for the record
Mally