List of questions
1. What is the smallest positive integer > 1 that has a
remainder of 1, when divided by each of 3,5 and 7?

2. How can I find a set of four integers such that the sum
of any two of them is a square.

Question # 1
What is the smallest positive integer > 1 that has a
remainder of 1, when divided by each of 3,5 and 7?

Let x be the name of the integer we are to find.
If x has a remainder of 1 when divided by 3,
then (x-1) has a remainder of 0 when divided by 3.
That is, (x-1) is a multiple of 3.

If x has a remainder of 1 when divided by 5,
then (x-1) is a multiple of 5.

If x has a remainder of 1 when divided by 7,
then (x-1) is a multiple of 7.

The smallest positive integer that is, at the same time,
a multiple of 3, a multiple of 5, and a multiple of 7,
is 3 * 5 * 7 = 105.

(x-1) = 105
x = 106.

106 is the smallest positive integer > 1 such that
the remainder is 1 when divided by any of 3,5, or 7.

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Question # 2
How can I find a set of four integers such that the sum of
any two of them is a square.

Choose 6 integers, t1,t2,t3,s1,s2,s3.
Define

a1 = real part of
((t1+s1+1)+i(t1-s1))((t2+s2+1)+i(t2-s2))((t3+s3+1)+i(t3-s3))

a2 = imaginary part of
((t1+s1+1)+i(t1-s1))((t2+s2+1)+i(t2-s2))((t3+s3+1)+i(t3-s3))

a3 = real part of
((t1+s1+1)+i(t1-s1))((t2+s2+1)+i(t2-s2))((t3+s3+1)-i(t3-s3))

a4 = imaginary part of
((t1+s1+1)+i(t1-s1))((t2+s2+1)+i(t2-s2))((t3+s3+1)-i(t3-s3))

a5 = real part of
((t1+s1+1)+i(t1-s1))((t2+s2+1)-i(t2-s2))((t3+s3+1)+i(t3-s3))

a6 = imaginary part of
((t1+s1+1)+i(t1-s1))((t2+s2+1)-i(t2-s2))((t3+s3+1)+i(t3-s3))

Define either
p1 = (a1**2 + a3**2 - a6**2)/2
p2 = (a1**2 + a6**2 - a3**2)/2
p3 = (a3**2 + a6**2 - a1**2)/2
p4 = (a5**2 + a4**2 - a1**2)/2
or
p1 = (a1**2 + a3**2 - a5**2)/2
p2 = (a1**2 + a5**2 - a3**2)/2
p3 = (a3**2 + a5**2 - a1**2)/2
p4 = (a6**2 + a4**2 - a1**2)/2

according to which will make p1,p2,p3,p4 to be integers.

p1 + p2 = a1**2
p1 + p3 = a3**2
p1 + p4 = a5**2
p2 + p3 = a6**2
p2 + p4 = a4**2
p3 + p4 = a2**2

or

p1 + p2 = a1**2
p1 + p3 = a3**2
p1 + p4 = a6**2
p2 + p3 = a5**2
p2 + p4 = a4**2
p3 + p4 = a2**2

Numerical Example
Choose t1 = 2, t2 = 3, t3 = 5, s1=1, s2=1,s3=1

(4+i)(5+2i)(7+4i)
= ((4+i)(5+2i))(7+4i)
= (18+13i)(7+4i)
= (74+163i)

a1 = 74
a2 = 163

(4+i)(5+2i)(7-4i)
= ((4+i)(5+2i))(7-4i)
= (18+13i)(7-4i)
= (178+19i)

a3 = 178
a4 = 19

(4+i)(5-2i)(7+4i)
= ((4+i)(5-2i))(7+4i)
= (22-3i)(7+4i)
= (166+67i)

a5 = 166
a6 = 67

a1**2 = 74**2 = 5476
a2**2 = 163**2 = 26569
a3**2 = 178**2 = 31684
a4**2 = 19**2 = 361
a5**2 = 166**2 = 27556
a6**2 = 67**2 = 4489

p1 = (a1**2 + a3**2 - a5**2)/2
p1 = (5476 + 31684 - 27556)/2
p1 = 4802

p2 = (a1**2 + a5**2 - a3**2)/2
p2 = (5476 + 27556 - 31684)/2
p2 = 674

p3 = (a3**2 + a5**2 - a1**2)/2
p3 = (31684 + 27556 - 5476)/2
p3 = 26882

p4 = (a6**2 + a4**2 - a1**2)/2
p4 = (4489 + 361 - 5476 )/2
p4 = -313

p1 + p2 = 5476 = 74**2
p1 + p3 = 31684 = 178**2
p1 + p4 = 4489 = 67**2
p2 + p3 = 27556 = 166**2
p2 + p4 = 361 = 19**2
p3 + p4 = 26569 = 163**2

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