Problems & Puzzles: Puzzles

Puzzle 483. p^2=a^2&b^2 JM Bergot posed this nice question: Are there more primes like 223 with 223^2 = 49,729, having digits that are both squares: 49=7^2 abd 729=27^2? C. Rivera found that the following smaller primes having this property are: 7, 13, 19 & 41, because: 7^2=49=2^2&3^2 13^2= 169=4^2&3^2 19^2= 361=6^2&1^2 41^2= 1681=4^2&9^2 The largest one found was this: 1012639687^2=1025439135687457969 = 320224786^2&3^2 Q1. Can you find larger examples? Q2 Can you find examples (other than  7^2=49=2^2&3^2) such that p, a & b are prime numbers? Q3: Is there a larger prime than 446653271 such that all the digits of p^2 are squares? 446653271^2= 199499144494999441 (Suggested model: p^2=a^2&b^2&c^2...z^2, a, b, ... z is 1 or 4 or 9) Note: This number (446653271) has been reported by Hisanori Mishima here. See my puzzle 313 too. Q4. Find larger primes such that p^2=a^2&b^2&c^2...z^2 where a, b, ... z contains not necessarily just one digit as in Q3. Example: 503609521^2 = 253622549641849441 = 5^2&6^2&15^2&7^2&8^2&43^2&21^2

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Contributions came from Qu Shungliang and JC Rosa.

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Qu Shungliang wrote:

Q1:
43438666913851579167519189742608957108310765136832211688777298879303209
55240657039793226965207918761503190378576593650917594297099740625226799
14706246333614228130639752669529770859607425539322819103794948807560224
59650933342339476019761413786579020507541574158596816047811468598374699
54413202905004255997860326042150820329781470371140858278787670052596776
152867604624102210307150131213^2=x^2 & 3^2.

43438666913851579167519189742608957108310765136832211688777298879303209
55240657039793226965207918761503190378576593650917594297099740625226799
14706246333614228130639752669529770859607425539322819103794948807560224
59650933342339476019761413786579020507541574158596816047811468598374699
54413202905004255997860326042150820329781470371140858278787670052596776
152867604624102210307150131213 (385 digits) is a prime.

we fix b=3, then p^2=a^2*10+9， is a pell equation.
Solve it, we get :
X0 = 7
Y0 = 2
Xn+1 = 19 Xn + 60 Yn
Yn+1 = 6 Xn + 19 Yn
We check Xn+1， if it’s a prime, we found a solution.

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JC Rosa wrote:

Q1: I found the following solution :
p=2161873163521 ; a=683644320912 ; b=1

2161873163521^2=683644320912^2&1^2

later he added:

About Q1 my best result is now:

p=1614724747739596175355052710959502435022061234407
a=510620799709794675713320084875894671585877391142

p^2=a^2&3^2

Q2: It is impossible to find one solution where p,a,b are primes
other than 7^2=2^2&3^2

proof : We have p^2=a^2&b^2
A) a=2 ; p^2=2^2&b^2
Let L=length of b^2
We have p^2=4*10^L+b^2 from where p^2-b^2=4*10^L
(p-b)(p+b)=4*10^L (1)
p and b are odd primes therefore (p-b) and (p+b) are even numbers.
Let p-b=2z ; p+b=2t where z and t have not the same parity ; gcd(z,t)=1 and
t>z ( thus p=t+z and b=t-z )
The equality (1) becomes : 4zt=4*10^L
zt=10^L
if L=1 then zt=10 , z=2 , t=5 from where p=2+5=7 ; b=5-2=3
7^2=2^2&3^2
if L>=2 then zt=10^L=2^L*5^L
from where z=2^L ; t=5^L and b=5^L-2^L=0 mod 3
therefore b is not prime
B) a>=3 ; a odd prime
+ if b=3 then a is an even number thus a is not a prime number

+ b>=7 and L>=2

With the same notations than above now we have :
zt=2^(L-2)*5^L*a^2 ( gcd (z,t)=1 )
In all cases ( z even , t odd , L even , L odd or z odd , t even , L even , L odd)
we obtain b=0 mod 3. Thus b is not a prime number.

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