Sorry, I mean numbers using all nine digits such as 201456789 for example. All permutations of nine digit numbers where each digit is used only once. Cases where the zero is first are 8 digit numbers.Mark44 said:Maybe I'm not understanding your question, but 1, 4, and 9 are in the set and all have integer square roots. If you're thinking something else, please clarify what you're doing.
The similar problem with digits 1...9 does have integer square roots. I think having a zero replace the number 3 restricts the possibilities.DaveC426913 said:What paths toward a solution does your "suspicion" lead you down?
I realize that but it's not a homework problem or assignment and I spent some time on it. I can do this, let the letters ##a,b,c,d,e## represent unknown digits then we have; $$(a,b,c,d,e)(a,b,c,d,e)=(a^2,2ab,2ac+b^2,2ad+2bc,2ae+2bd+c^2,2be+2cd,2ce+d^2,2de,e^2)$$ maybe I can use logic to eliminate possibilities. For example, if ##e=0##, there will be two zero's in the answer which is forbidden. If any combination of candidate digits yield a 3 that combination is forbidden. I can see there are a lot of rules here like ##a≠9## which has too many digits, ##a≠e## ect...DaveC426913 said:OK, it's just that it's not the PF way to start handing out answers without the poster first showing their own attempts and/or research. But that's just my take.
But don't the numbers you're asking about have 9 digits (including a possible leading 0 digit)? Why are you using only 5 digits?bob012345 said:I realize that but it's not a homework problem or assignment and I spent some time on it. I can do this, let the letters ##a,b,c,d,e## represent unknown digits then we have; $$(a,b,c,d,e)(a,b,c,d,e)=(a^2,2ab,2ac+b^2,2ad+2bc,2ae+2bd+c^2,2be+2cd,2ce+d^2,2de,e^2)$$
Nit -- the abbreviation for et cetera is etc., not ect.bob012345 said:maybe I can use logic to eliminate possibilities. For example, if ##e=0##, there will be two zero's in the answer which is forbidden. If any combination of candidate digits yield a 3 that combination is forbidden. I can see there are a lot of rules here like ##a≠9## which has too many digits, ##a≠e## ect...
We are trying to determine if a nine digit number with the restrictions above can have an integer square root. For example the number 102456789 has square root 10122.094..... so it doesn't work. Also squaring 10121 gives 102434641 which doesn't work because it repeats digits 1,4 and has a 3 as well which violate my rules. I only need one counterexample to disprove my suspicion. The only rule on the square root itself is that it be an integer and it has to have five digits to make a nine digit square.Mark44 said:But don't the numbers you're asking about have 9 digits (including a possible leading 0 digit)? Why are you using only 5 digits?Nit -- the abbreviation for et cetera is etc., not ect.
I can see how you came up with 102456789, but you do realize that there are 9! = 362,880 different permutation of the digits in your set, right? Do you plan to check them all by what seems to be a brute force approach?bob012345 said:For example the number 102456789 has square root 10122.094.
Actually a lot can be eliminated. So far, I got it down to 21,623 possibilities. Those are the only possible integer roots that will make a nine number square.Mark44 said:I can see how you came up with 102456789, but you do realize that there are 9! = 362,880 different permutation of the digits in your set, right? Do you plan to check them all by what seems to be a brute force approach?
You can check all the possibilities quickly if you turn it around and square all integers between the lower and upper possible limits. I used Python to check there are none. I checked for both 8-digit numbers that were a permutation of the digits 1-9, except 3. And for 9-digit numbers, whose digits were a permutation of the digits 0-9, except 3. These programs took less than a second to run.bob012345 said:Actually a lot can be eliminated. So far, I got it down to 21,623 possibilities. Those are the only possible integer roots that will make a nine number square.
check_str = ['1', '2','3', '4', '5', '6', '7', '8', '9']
#
for k in range(10117, 31427):
k2 = str(k**2)
sorted_k2 = sorted(k2)
#
if sorted_k2 == check_str:
print(k, k**2)Thanks. I used the upper and lower limits to reduce the number of possibilities as well as logic. For example there are only 21,623 possibilities based on limits but that can be reduced to 7,344 by noticing the values of the digits of the possible roots are constrained as well in various ways. Ramanujan would probably have just looked at it and said why it's impossible.PeroK said:You can check all the possibilities quickly if you turn it around and square all integers between the lower and upper possible limits. I used Python to check there are none. I checked for both 8-digit numbers that were a permutation of the digits 1-9, except 3. And for 9-digit numbers, whose digits were a permutation of the digits 0-9, except 3. These programs took less than a second to run.
To check the code worked, I looked for integers whose square was a permutation of the digits 1-9. The lowest possible integer is 11, 112 (which is approx ##\sqrt{123456789}##) and the highest is 31427 (which is approx ##\sqrt{9
That's not hard to prove. A number is divisible by 3 iff the sum of its digits is divisible by 3. And divisible by 9 iff the sum of its digits is divisible by 9. This follows from 10 being 1 modulo 3 and 9.bob012345 said:A friend gave me a rule that seems to work. If the sum of the digits is divisible by 3 but not by 9, there are no permutations of those digits that make a perfect square. If it's not divisible by either, it's indeterminate. If it's divisible by both, it's indeterminate.
Is that New Math?PeroK said:This follows from 10 being 1 modulo 3 and 9.
PeroK said:This follows from 10 being 1 modulo 3 and 9.
No, it's number theory, which has been around for a lot longer than so-called "New Math."bob012345 said:Is that New Math?
It's not an original result, I'm sorry to say.bob012345 said:Is that New Math?
'New Math' was an American term from the 1960's when the U.S. changed the math teaching style and the new style confounded and confused both teachers and parents.PeroK said:It's not an original result, I'm sorry to say.
The integer square roots of {0,1,2,4,5,6,7,8,9} are as follows:
√0 = 0, √1 = 1, √2 is not an integer, √4 = 2, √5 is not an integer, √6 is not an integer, √7 is not an integer, √8 = 2, √9 = 3.
Some numbers do not have integer square roots because their square roots result in decimal numbers or irrational numbers. In the case of {2,5,6,7}, their square roots are not whole numbers.
To calculate the integer square root of a number, you can use methods such as prime factorization, trial and error, or using a calculator. For example, to find the square root of 9, you would know that 3 x 3 = 9, so the square root of 9 is 3.
Integer square roots are important in mathematics as they help in simplifying calculations and finding whole number solutions to equations. They are also used in various mathematical concepts and applications.
Yes, there are certain patterns and rules for finding integer square roots. For example, perfect squares like 1, 4, 9, 16, 25, etc., have integer square roots that are whole numbers. Understanding these patterns can help in quickly determining the integer square roots of numbers.