601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691.
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Square numbers in that range are 625 and 676.
Sixteen of them.
16 prime numbers
The prime numbers between 700 and 800 are: 701 709 719 727 733 739 743 751 757 761 769.
The number halfway between them is 600.